chapter 4 continuous random variable.pdf

7/26/2019 Chapter 4 Continuous Random Variable.pdf

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Chapter 4: Continuous Random

Variable

- Normal Distribution

BENG 2142 Statistics

Dr. Rahifa binti Ranom


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4.1 Definition

A Continuous random variable is a random

variable where the data can take infinitely many

values.

Continuous RV deals with data in interval sets.

Examples:

Time taken for something to be done

Weight of students in a class Length of machine parts




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4.2 Probability Density Function (pdf)

A function that gives the probability of a continuous randomvariable to take on a given value (in a range/interval).

Also known as probability function or probability distribution of

the continuous random variable X

Properties:

1)0 12)

= 13) < < = Note: For Continuous RV,

< < < <




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Example 4.1

Consider the probability density function

= , 0 < < 1, , 1 < 2,0, elsewhere.a) Find

.

b) Evaluate ( < 1.2).c) Evaluate (0.5 < < 1).(Ans: 2; 0.68; 0.375)




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4.3 Cumulative Distribution Function(CDF) of a

Continuous RV

Cumulative distribution function(CDF) - of a continuousrandom variable with probability distribution function

is given by

= = , < < .Hence,

< < = and ()




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Example 4.2

Consider the probability density function

= , 0 < < 1,0, elsewhere.a) Evaluate .b) Find () and use it to evaluate (0.3 < < 0.6).

(Ans: 3/2; = 0, < 0,/, 0 < 11, 1. ; 0.3004)




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4.6 Normal Distribution

The most important continuous probability

distribution in the entire field of statistics.

The graph normal curve bell shape.

Normally used in physical measurement areas;

ex: meteorological experiments, rainfall studies,

measurements of manufactured parts




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4.6 Normal Distribution (cont.) 17th century byAbraham De Moivre

A continuous random variable having thefollowing bell-shaped distribution is called anormal random variable.



NOTE:

Total area under the

curve is 1.0

The curve is symmetric

about the mean

The two tails of the

curve extend indefinitely


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4.6 Normal Distribution (cont.)

The probability density function (pdf) of thenormal random variable X, with mean andvariance , is = 12

()

, < < The probability of the random variable Xbetween

= and

= equals area under the curve

bounded by the two coordinates = and =

1 ()




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4.6 Normal Distribution (cont.)

For normal curves with different means

and variances:

It will be hard to find the area under the

bounded region.

What we do?

Transform all the observations of any

normal random variableXto a new set ofobservations of a normal random variable

Zwith mean 0 and variance 1. We called

this as standard normal distribution.




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4.7 Standard Normal Distribution

The transformation: = IfXfalls between = and = ,thenZwill fall between

=

and

= The standard normal distribution is a special case

of the normal distribution, with

(a) The mean,

= 0;

(b) The variance, = 1(c) The units of the standard normal distributioncurve are denoted byz, called asz-values or

z-scores.




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4.7 Standard Normal Distribution (cont.)

Transformation of normal to standard normal rv:

< < = 12

()

=12

= < <




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Example 4.3

1) Given a standard normal distribution, find the areaunder the curve that lies

a) To the right of = 1.84 andb) Between

= 1.97and

= 0.86.

(Ans: 0.0329; 0.7807)

2) Given a standard normal distribution, evaluate

a)

( < 3.25)b) 1.7 < < 2.5c) > 2.75d) 1.37 < < 0

(Ans: 0.0006; 0.0384; 0.997; 0.4147)



i i


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Example 4.3

3) Given a standard normal distribution, find the value

of k such thata) < = 0.0427;b) > = 0.2946;c)

0.93 < < = 0.7235.

(Ans: -1.72; 0.54; 1.28)

4) Given the normally distributed variable Xwith mean

18 and standard deviation 2.5, find

a)

( < 15)b) The value of ksuch that < = 0.2236;c) The value of ksuch that > = 0.1814;d)(17 < < 21)

(Ans: 0.1151; 16.1; 20.275; 0.5403)



BENG 2142 S i i


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Example 4.3

5) The loaves of rye bread distributed to local storesby a certain bakery have an average length of 30

centimeters and a standard deviation of 2

centimeters. Assuming that the lengths are

normally distributed, what percentage of theloaves are

a) Longer than 31.7 centimeters?

b) Between 29.3 and 33.5 centimeters in length?

c) Shorter than 25.5 centimeters?

(Ans: 19.77%; 59.67%; 1.22%)



BENG 2142 St ti ti


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Example 4.3

6) A stamping machine produces can tops whosediameters are normally distributed with a

standard deviation of 0.01 inch. At what normal

(mean) diameter should the machine be set so

that no more than 5% of the can tops producedhave diameters exceeding 3 inches?

(Ans: 2.9836)



BENG 2142 Statisti s


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4.8 Normal Approximation to the Binomial

If X is a binomial random variable with large n ( 30),with = and = ,the distribution approximately follows Normal

distribution = Continuity correction:( ) 0.5 <


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Example 4.4

1) If 20% of the memory chips made in a certainplant are defective, what are the probabilities

that in a lot of 100 randomly chosen for

inspection,

a) At most 15 will be defective?b) Exactly 15 will be defective?

(Ans: 0.1292; 0.0454)

2) The probability that a patients recovers from a

rare blood disease is 0.4. If 100 people are known

to have contracted this disease, what is the

probability that fewer than 30 survive?

(Ans: 0.0162)



chapter 4 continuous random variable.pdf

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