1 some continuous probability distributions uniform, normal, exponential, gamma and chi-square...

86
1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

Upload: susanna-carr

Post on 30-Dec-2015

249 views

Category:

Documents


1 download

TRANSCRIPT

Page 1: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

1

SOME CONTINUOUS PROBABILITY

DISTRIBUTIONSUniform, Normal, Exponential,

Gamma and Chi-Square Distributions

Page 2: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

2

– A random variable X is said to be uniformly distributed if its density function is

– The expected value and the variance are

12)ab(

)X(V2

baE(X)

.bxaab

1)x(f

2

Uniform Distribution

Page 3: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

Indicator functions

• It is sometimes convenient to express the p.m.f. or p.d.f. by using indicator functions. This is especially true when the range of random variable depends on a parameter.

• Ex: Uniform distribution

3

Page 4: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

4

• Example 1– The daily sale of gasoline is uniformly distributed

between 2,000 and 5,000 gallons. Find the probability that sales are:

– Between 2,500 and 3,000 gallons

– More than 4,000 gallons

– Exactly 2,500 gallons

2000 5000

1/3000

f(x) = 1/(5000-2000) = 1/3000 for x: [2000,5000]

x2500 3000

P(2500X3000) = (3000-2500)(1/3000) = .1667

Uniform Distribution

Page 5: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

5

• Example 1– The daily sale of gasoline is uniformly distributed

between 2,000 and 5,000 gallons. Find the probability that sales are:

– Between 2,500 and 3,500 gallons

– More than 4,000 gallons

– Exactly 2,500 gallons

2000 5000

1/3000

f(x) = 1/(5000-2000) = 1/3000 for x: [2000,5000]

x4000

P(X4000) = (5000-4000)(1/3000) = .333

Uniform Distribution

Page 6: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

6

• Example 1– The daily sale of gasoline is uniformly distributed

between 2,000 and 5,000 gallons. Find the probability that sales are:

– Between 2,500 and 3,500 gallons

– More than 4,000 gallons

– Exactly 2,500 gallons

2000 5000

1/3000

f(x) = 1/(5000-2000) = 1/3000 for x: [2000,5000]

x2500

P(X=2500) = (2500-2500)(1/3000) = 0

Uniform Distribution

Page 7: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

7

Normal Distribution

• This is the most popular continuous distribution.

– Many distributions can be approximated by a normal distribution.

– The normal distribution is the cornerstone distribution of statistical inference.

Page 8: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

8

• A random variable X with mean and variance is normally distributed if its probability density function is given by

...71828.2eand...14159.3where

0;xe2

1)x(f

2x)2/1(

...71828.2eand...14159.3where

0;xe2

1)x(f

2x)2/1(

Normal Distribution

Page 9: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

The Shape of the Normal Distribution

The normal distribution is bell shaped, and symmetrical around

Why symmetrical? Let = 100. Suppose x = 110.

2210

)2/1(100110

)2/1(e

21

e2

1)110(f

Now suppose x = 90210

)2/1(210090

)2/1(e

2

1e

2

1)90(f

11090

Page 10: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

10

The Effects of and

How does the standard deviation affect the shape of f(x)?

= 2

=3 =4

= 10 = 11 = 12How does the expected value affect the location of f(x)?

Page 11: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

11

• Two facts help calculate normal probabilities:– The normal distribution is symmetrical.– Any normal distribution can be transformed into

a specific normal distribution called…

“STANDARD NORMAL DISTRIBUTION”

Example

The amount of time it takes to assemble a computer is normally distributed, with a mean of 50 minutes and a standard deviation of 10 minutes. What is the probability that a computer is assembled in a time between 45 and 60 minutes?

Finding Normal Probabilities

Page 12: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

12

STANDARD NORMAL DISTRIBUTION

• NORMAL DISTRIBUTION WITH MEAN 0 AND VARIANCE 1.

• IF X~N( , 2), THEN

NOTE: Z IS KNOWN AS Z SCORES.

• “ ~ “ MEANS “DISTRIBUTED AS”

~ (0,1)X

Z N

Page 13: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

13

• Solution– If X denotes the assembly time of a computer, we

seek the probability P(45<X<60).– This probability can be calculated by creating a new

normal variable the standard normal variable.

x

xXZ

x

xXZ

E(Z) = = 0 V(Z) = 2 = 1

Every normal variablewith some and, canbe transformed into this Z.

Therefore, once probabilities for Zare calculated, probabilities of any normal variable can be found.

Finding Normal Probabilities

Page 14: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

Standard normal probabilities

14

Copied from Walck, C (2007) Handbook on Statistical Distributions for experimentalists

Page 15: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

Standard normal table 1

15

Page 16: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

Standard normal table 2

16

Page 17: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

17

Standard normal table 3

Page 18: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

18

• Example - continued

P(45<X<60) = P( < < )45 X 60- 50 - 5010 10

= P(-0.5 < Z < 1)

To complete the calculation we need to compute the probability under the standard normal distribution

Finding Normal Probabilities

Page 19: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

19

z 0 0.01 ……. 0.05 0.060.0 0.0000 0.0040 0.0199 0.02390.1 0.0398 0.0438 0.0596 0.0636

. . . . .

. . . . .1.0 0.3413 0.3438 0.3531 0.3554

. . . . .

. . . . .1.2 0.3849 0.3869 ……. 0.3944 0.3962

. . . . . .

. . . . . .

Standard normal probabilities have been calculated and are provided in a table .

The tabulated probabilities correspondto the area between Z=0 and some Z = z0 >0 Z = 0 Z = z0

P(0<Z<z0)

Using the Standard Normal Table

Page 20: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

20

• Example - continued

P(45<X<60) = P( < < )45 X 60- 50 - 5010 10

= P(-.5 < Z < 1)

z0 = 1z0 = -.5

We need to find the shaded area

Finding Normal Probabilities

Page 21: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

21

P(-.5<Z<0)+ P(0<Z<1)

P(45<X<60) = P( < < )45 X 60- 50 - 5010 10

z 0 0.01 ……. 0.05 0.060.0 0.0000 0.0040 0.0199 0.02390.1 0.0398 0.0438 0.0596 0.636. . . . .. . . . .

1.0 0.3413 0.3438 0.3531 0.3554. . . . .

P(0<Z<1

• Example - continued

= P(-.5<Z<1) =

z=0 z0 = 1z0 =-.5

.3413

Finding Normal Probabilities

Page 22: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

22

• The symmetry of the normal distribution makes it possible to calculate probabilities for negative values of Z using the table as follows:

-z0 +z00

P(-z0<Z<0) = P(0<Z<z0)

Finding Normal Probabilities

Page 23: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

23

z 0 0.01 ……. 0.05 0.060.0 0.0000 0.0040 0.0199 0.02390.1 0.0398 0.0438 0.0596 0.636

. . . . .

. . . . .0.5 0.1915 …. …. ….

. . . . .

• Example - continued

Finding Normal Probabilities

.3413

.5-.5

.1915

Page 24: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

24

z 0 0.1 ……. 0.05 0.060.0 0.0000 0.0040 0.0199 0.02390.1 0.0398 0.0438 0.0596 0.636

. . . . .

. . . . .0.5 0.1915 …. …. ….

. . . . .

• Example - continued

Finding Normal Probabilities

.1915.1915.1915.1915

.3413

.5-.5

P(-.5<Z<1) = P(-.5<Z<0)+ P(0<Z<1) = .1915 + .3413 = .5328

1.0

Page 25: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

25

z 0 0.01 ……. 0.05 0.060.0 0.5000 0.5040 0.5199 0.52390.1 0.5398 0.5438 0.5596 0.5636

. . . . .

. . . . .0.5 0.6915 …. …. ….

. . . . .

• Example - continued

Finding Normal Probabilities

.3413

P(Z<-0.5)=1-P(Z>-0.5)=1-0.6915=0.3085By Symmetry

P(Z<0.5)

This table provides probabilities from -∞ to z0

Page 26: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

26

10%0%

20-2

(i) P(X< 0 ) = P(Z< ) = P(Z< - 2)0 - 10

5

=P(Z>2) =Z

X

• Example– The rate of return (X) on an investment is normally distributed

with a mean of 10% and standard deviation of (i) 5%, (ii) 10%.

– What is the probability of losing money?

.4772

0.5 - P(0<Z<2) = 0.5 - .4772 = .0228

Finding Normal Probabilities

Page 27: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

27

10%0%

-1

(ii) P(X< 0 ) = P(Z< ) 0 - 10

10

= P(Z< - 1) = P(Z>1) =Z

X

• Example– The rate of return (X) on an investment is normally

distributed with mean of 10% and standard deviation of (i) 5%, (ii) 10%.

– What is the probability of losing money?

.3413

0.5 - P(0<Z<1) = 0.5 - .3413 = .1587

Finding Normal Probabilities

1

Page 28: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

28

AREAS UNDER THE STANDARD NORMAL DENSITY

P(0<Z<1)=.3413

Z0 1

Page 29: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

29

AREAS UNDER THE STANDARD NORMAL DENSITY

.3413 .4772

P(1<Z<2)=.4772-.3413=.1359

Page 30: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

30

EXAMPLES

• P( Z < 0.94 ) = 0.5 + P( 0 < Z < 0.94 )

= 0.5 + 0.3264 = 0.8264

0.940

0.8264

Page 31: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

31

EXAMPLES

• P( Z > 1.76 ) = 0.5 – P( 0 < Z < 1.76 )

= 0.5 – 0.4608 = 0.0392

1.760

0.0392

Page 32: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

32

EXAMPLES

• P( -1.56 < Z < 2.13 ) =

= P( -1.56 < Z < 0 ) + P( 0 < Z < 2.13 )

= 0.4406 + 0.4834 = 0.9240

P(0 < Z < 1.56)

-1.56 2.13

0.9240

Because of symmetry

Page 33: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

33

STANDARDIZATION FORMULA

• If X~N( , 2), then the standardized value Z of any ‘X-score’ associated with calculating probabilities for the X distribution is:

• The standardized value Z of any ‘X-score’ associated with calculating probabilities for the X distribution is:

(Converse Formula)

XZ

.x z

Page 34: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

34

• Sometimes we need to find the value of Z for a given probability

• We use the notation zA to express a Z value for which P(Z > zA) = A

Finding Values of Z

zA

A

Page 35: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

35

PERCENTILE• The pth percentile of a set of measurements

is the value for which at most p% of the measurements are less than that value.

• 80th percentile means P( Z < a ) = 0.80

• If Z ~ N(0,1) and A is any probability, then

P( Z > zA) = A

A

zA

Page 36: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

36

• Example – Determine z exceeded by 5% of the population– Determine z such that 5% of the population is below

• Solutionz.05 is defined as the z value for which the area on its right

under the standard normal curve is .05.

0.05

Z0.050

0.45

1.645

Finding Values of Z

0.05

-Z0.05

Page 37: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

37

EXAMPLES

• Let X be rate of return on a proposed investment. Mean is 0.30 and standard deviation is 0.1.

a) P(X>.55)=?b) P(X<.22)=?c) P(.25<X<.35)=?d) 80th Percentile of X is?e) 30th Percentile of X is?

Standardization formula

Converse Formula

Page 38: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

38

ANSWERSa)

b)

c)

d)

e)

X - 0.3 0.55 - 0.3P(X 0.55) P = Z > = 2.5 = 0.5 - 0.4938 = 0.0062

0.1 0.1

X - 0.3 0.22 - 0.3P(X 0.22) P = Z = 0.8 = 0.5 - 0.2881 = 0.2119

0.1 0.1

0.25 0.3 X - 0.3 0.35 - 0.3P(0.25 X 0.35) P 0.5 = Z = 0.5

0.1 0.1 0.1

= 2.*(0.1915) 0.3830

80th Percentile of X is 0.20. .3+(.85)*(.1)=.385x z

30th Percentile of X is 0.70. .3+(-.53)*(.1)=.247x z

Page 39: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

39

The Normal Approximation to the Binomial Distribution

• The normal distribution provides a close approximation to the Binomial distribution when n (number of trials) is large and p (success probability) is close to 0.5.

• The approximation is used only when

np 5 and

n(1-p) 5

Page 40: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

40

The Normal Approximation to the Binomial Distribution

• If the assumptions are satisfied, the Binomial random variable X can be approximated by normal distribution with mean = np and 2 = np(1-p).

• In probability calculations, the continuity correction improves the results. For example, if X is Binomial random variable, then

P(X a) ≈ P(X<a+0.5)

P(X a) ≈ P(X>a-0.5)

Page 41: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

41

EXAMPLE• Let X ~ Binomial(25,0.6), and want to find P(X ≤ 13).

• Exact binomial calculation:

• Normal approximation (w/o correction): Y~N(15,2.45²)

13

0x

x25x 267.0)4.0()6.0(x

25)13X(P

206.0)82.0Z(P)45.2

1513Z(P)13Y(P)13X(P

Normal approximation is good, but not great!

Page 42: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

EXAMPLE, cont.

42Copied from Casella & Berger (1990)

Bars – Bin(25,0.6); line – N(15, 2.45²)

Page 43: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

EXAMPLE, cont.

• Normal approximation (w correction):

Y~N(15,2.45²)

43

271.0)61.0Z(P)5.13Y(P)5.13X(P)13X(P

Much better approximation to the exact value: 0.267

Page 44: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

44

Exponential Distribution• The exponential distribution can be used to model

– the length of time between telephone calls– the length of time between arrivals at a service station– the lifetime of electronic components.

• When the number of occurrences of an event follows the Poisson distribution, the time between occurrences follows the exponential distribution.

Page 45: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

45

A random variable is exponentially distributed if its probability density function is given by

f(x) = e-x, x>=0.is the distribution parameter >0).

E(X) = V(X) = 2

Exponential Distribution

The cumulative distribution function isF(x) =1e-x/, x0

/1, 0, 0x

Xf x e x

is a distribution parameter.

Page 46: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

46

0

0.5

1

1.5

2

2.5f(x) = 2e-2x

f(x) = 1e-1x

f(x) = .5e-.5x

0 1 2 3 4 5

Exponential distribution for = .5, 1, 2

0

0.5

1

1.5

2

2.5

a bP(a<X<b) = e-a/ e-b/

Page 47: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

47

• Finding exponential probabilities is relatively easy:– P(X < a) = P(X ≤ a)=F(a)=1 – e –a/

– P(X > a) = e–a/

– P(a< X < b) = e – a/ – e – b/

Exponential Distribution

Page 48: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

48

• Example– The lifetime of an alkaline battery is

exponentially distributed with mean 20 hours.– Find the following probabilities:

• The battery will last between 10 and 15 hours.• The battery will last for more than 20 hours.

Exponential Distribution

Page 49: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

49

• Solution– The mean = standard deviation =

20 hours.– Let X denote the lifetime.

• P(10<X<15) = e-.05(10) – e-.05(15) = .1341• P(X > 20) = e-.05(20) = .3679

Exponential Distribution

Page 50: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

50

• Example The service rate at a supermarket checkout

is 6 customers per hour. – If the service time is exponential, find the

following probabilities:• A service is completed in 5 minutes,• A customer leaves the counter more than 10

minutes after arriving• A service is completed between 5 and 8 minutes.

Exponential Distribution

Page 51: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

51

• Solution– A service rate of 6 per hour =

A service rate of .1 per minute ( = .1/minute).– P(X < 5) = 1-e-.lx = 1 – e-.1(5) = .3935– P(X >10) = e-.lx = e-.1(10) = .3679– P(5 < X < 8) = e-.1(5) – e-.1(8) = .1572

Exponential Distribution

Page 52: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

52

Exponential Distribution

• If pdf of lifetime of fluorescent lamp is exponential with mean 0.10, find the life for 95% reliability?

The reliability function = R(t) = 1F(t) = e-t/

R(t) = 0.95 t = ?

Page 53: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

53

• X~ Gamma(,)

GAMMA DISTRIBUTION

1 /1, 0, 0, 0xf x x e x

2 and E X Var X

11 , M t t t

Page 54: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

54

GAMMA DISTRIBUTION

• Gamma Function:

1

0

xx e dx

where is a positive integer.

Properties:

1 , 0

1 ! for any integer 1n n n 12

Page 55: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

55

)()(

)(

)1(

)()1(

0

1

0

1

0

dxexx

dxexxdxex

x

xx

GAMMA DISTRIBUTION

since the last integral is the expectation of a Gamma distribution with parameters alpha and 1.

Page 56: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

56

• Let X1,X2,…,Xn be independent rvs with Xi~Gamma(i, ). Then,

GAMMA DISTRIBUTION

1 1

~ ,n n

i ii i

X Gamma

•Let X be an rv with X~Gamma(, ). Then,

~ , where is positive constant.cX Gamma c c • Let X1,X2,…,Xn be a random sample with Xi~Gamma(, ). Then,

1 ~ ,

n

ii

XX Gamma n

n n

Page 57: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

GAMMA DISTRIBUTION

• Special cases: Suppose X~Gamma(α,β)– If α=1, then X~ Exponential(β)– If α=p/2, β=2, then X~ 2 (p) (will come back in a min.)

– If Y=1/X, then Y ~ inverted gamma.

• Gamma approximates to Normal distribution on the limit as alpha goes to infinity.

57

Page 58: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

Integral tricks

• Recall: h(t) is an odd function if h(-t)=-h(t);

It is an even function if h(-t)=h(t).

• Ex: h(x)=x exp{-x²/2} odd;

h(x)= exp{-x²/2-x} neither odd nor even

• odd function =0

• even function = 2* even function58

0

Page 59: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

59

CHI-SQUARE DISTRIBUTION

• X~ 2()= Gamma(/2,2)

and 2E X Var X

Chi-square with degrees of freedom

2/1)21()( 2/ tttM

,...2,1,0,)2/(2

1)( 2/12/

2/

xexxf x

Page 60: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

60

DEGREES OF FREEDOM• In statistics, the phrase degrees of freedom is

used to describe the number of values in the final calculation of a statistic that are free to vary.

• The number of independent pieces of information that go into the estimate of a parameter is called the degrees of freedom (df) .

• How many components need to be known before the vector is fully determined?

Page 61: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

• Chi-square (2) ≡ Exponential (2)

61

CHI-SQUARE DISTRIBUTION

Page 62: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

62

CHI-SQUARE DISTRIBUTION

• If rv X has Gamma(,) distribution, then Y=2X/ has Gamma(,2) distribution. If 2 is positive integer, then Y has

distribution.

2

2

•Let X1,X2,…,Xn be a r.s. with Xi~N(0,1). Then,2 2

1

~n

i ni

X

•Let X be an rv with X~N(0, 1). Then,2 2

1~X

Page 63: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

63

BETA DISTRIBUTION

• The Beta family of distributions is a continuous family on (0,1) and often used to model proportions.

1111 , 0 1, 0, 0.

,f x x x x

B

where

2 and

1E X Var X

)(

)()()1(),(

1

0

11

dxxxB

Page 64: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

64

WEIBULL DISTRIBUTION• To model the failure time data or hazard

functions.• Hazard function:

Rate of change in prob that subject survives a little

past time t, given that subject survives upto time t.

• If X~Exp(), then Y=X1/ has Weibull(, ) distribution.

1 / , 0, 0, 0y

Yf y y e y

)tT|tTt(Pmil)t(h

0

Page 65: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

65

CAUCHY DISTRIBUTION• It is a symmetric and bell-shaped

distribution on (,) with pdf

E X Since , the mean does not exist.

• The mgf does not exist.• measures the center of the distribution and it is the median.

• If X and Y have N(0,1) distribution, then Z=X/Y has a Cauchy distribution with =0 and σ=1.

0,)

x(1

11)x(f

2

Page 66: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

66

CAUCHY DISTRIBUTION

• When studying hypothesis tests that assume normality, seeing how the tests perform on data from a Cauchy distribution is a good indicator of how sensitive the tests are to heavy-tail departures from normality.

• Likewise, it is a good check for robust techniques that are designed to work well under a wide variety of distributional assumptions.

Page 67: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

67

LOG-NORMAL DISTRIBUTION

• An rv X is said to have the lognormal distribution, with parameters µ and 2, if Y=ln(X) has the N(µ, 2) distribution.

•The lognormal distribution is used to model continuous random quantities when the distribution is believed to be skewed, such as certain income and lifetime variables.

Page 68: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

68

STUDENT’S T DISTRIBUTION

• This distribution will arise in the study of population mean when the underlying distribution is normal.

• Let Z be a standard normal rv and let U be a chi-square distributed rv independent of Z, with degrees of freedom. Then,

~/

ZX t

U

When n, XN(0,1).

Page 69: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

69

F DISTRIBUTION

• Let U and V be independent rvs with chi-square distributions with 1 and 2 degrees of freedom. Then,

1 2

1

,

2

/~

/U

X FV

Page 70: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

MULTIVARIATE

DISTRIBUTIONS

70

Page 71: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

BIVARIATE NORMAL DISTRIBUTION

• A pair of continuous rvs X and Y is said to have a bivariate normal distribution if it has a joint pdf of the form

71

2

2

y

2

y

1

x2

1

x22

21

yyx2

x

12

1exp

12

1y,xf

.11,0,0,y,x 21

Page 72: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

BIVARIATE NORMAL DISTRIBUTION

72

If ,,,,BVN~Y,X yx

2

2

2

1

2

2

2

1 ,N~Y and ,N~X yx

, then

and is the correlation coefficient btw X and Y.1. Conditional on X=x,

22

2X1

2y 1),x(N~xY

2. Conditional on Y=y,

22

1Y2

1x 1),y(N~yX

Page 73: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

Mixture of distributions

• Let f1(y) and f2(y) be density functions, and let a be a constant such that 0≤ a ≤1. Consider the function

f(y)=a f1(y) + (1-a) f2(y)

Such a density function is often called a mixture distribution. Here are some examples of real life applications for such distributions.

73

Page 74: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

Mixture of distributions

• Example 1: Financial returns often behave differently in normal situations and during crisis times. A mixture of two normal distributions with different means and variances can be assumed for returns, one for the returns during normal situations, and another for during crises.

74

Page 75: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

Mixture of distributions

• Example 2: The prices of houses in a particular neighborhood will tend to be similar, while prices in a different neighborhood may be extremely different. For instance, if we are to collect prices in both Cukurambar and Sincan, we will need a mixture of two distributions to describe these prices.

75

Page 76: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

Mixture of distributions

• Note that f(y) is a valid density function.

• Suppose that Y1 is a random variable with density function f1(y), and E(Y1)=μ1 and Var(Y1)= . Similarly, suppose that Y2 is a random variable with density function f2(y), and E(Y2)=μ2 and Var(Y2)= . Assume that Y is random variable whose density is a mixture of the densities corresponding to Y1 and Y2.

i) It can be shown that E(Y)= a μ1 + (1-a) μ2

ii) It can be shown that

Var(Y)= a + (1-a) + a(1-a) (μ1 - μ2)2

76

21

22

21 2

2

Page 77: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

Problems

1. Mensa (from the Latin word for “mind”) is an international society devoted to intellectual pursuits. Any person who has an IQ in the upper 2% of the general population is eligible to join. If we assume that IQs are normally distributed with μ = 100 and σ = 16, what is the lowest IQ that will qualify a person for membership?

77

Page 78: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

Solution

78

Page 79: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

Problems

2. Suppose that numerical grades in a statistics class are values of a r.v. X which is Normally distributed with mean μ = 65 and standard deviation 15. Suppose that letter grades are assigned according to the following rule: student receives an A if X ≥ 85; B if 70 ≤ X < 85; C if 55 ≤ X < 70; D if 45 ≤ X < 55; and F if X ≤ 45. If a student is chosen at random from this class, calculate the probability that the student will earn i) A; ii) B; iii) F.

79

Page 80: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

Solution

80

Page 81: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

Problems

3. Economic conditions cause fluctuations in the prices of raw commodities as well as in finished products. Let X denote the price paid for a barrel of crude oil by the initial carrier, and let Y denote the price paid by the refinery purchasing the product from the carrier. Assume that the joint density for (X,Y) is given by

f(x,y)=c 20< x < y < 4081

Page 82: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

Problems

a) Find the value of c that makes this a joint density for a two-dimensional random variable.

b) Find the probability that the carrier will pay at least $25 per barrel and the refinery will pay at most $30 per barrel for the oil.

c) Find the probability that the price paid by the refinery exceeds that of the carrier by at least $10 per barrel.

82

Page 83: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

Problems

d) Are X and Y independent? Explain.

e) Find E(Y-X). Interpret this expectation in a practical sense.

83

Page 84: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

Solution

84

Page 85: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

Solution

85

Page 86: 1 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform, Normal, Exponential, Gamma and Chi-Square Distributions

Solution

d)

e)

86