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PROBABILITY DISTRIBUTIONS
David M. Lane. et al. Introduction to Statistics : Chapter 9
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 1 / 18
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Contents
1 Probability density, distribution
2 Probability distributions
Binomial distribution
Multinomial distribution
Normal distribution
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 2 / 18
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Next section
1 Probability density, distribution
2 Probability distributions
Binomial distribution
Multinomial distribution
Normal distribution
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 3 / 18
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Probability densityDe�nition
A probability density function or density of a continuous random variable, is a function thatdescribes the relative likelihood for this random variable to take on a given value.
The analogue for discrete random variable:
De�nition
A probability mass function is a function that gives the probability that a discrete randomvariable is exactly equal to some value.
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 4 / 18
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Probability as area under density curve
Any continuous probability distribution, areas under the density curve representprobabilities.
For any numbers a and b, the probability P(a< x < b) equals the area under the curvebetween a and b;
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 5 / 18
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Next section
1 Probability density, distribution
2 Probability distributions
Binomial distribution
Multinomial distribution
Normal distribution
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 6 / 18
ioc.pdf
Next subsection
1 Probability density, distribution
2 Probability distributions
Binomial distribution
Multinomial distribution
Normal distribution
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 7 / 18
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Bernoulli trials
Independent repeated trials of an experiment with exactly two possibleoutcomes are called Bernoulli trials.
Call one of the outcomes "success" and the other outcome "failure".Let p be the probability of success in a Bernoulli trial, and q = 1−p bethe probability of failure: "success" and "failure" are mutually exclusiveand exhaustive.
For example, a series of n trials consisting of m "successes" (event A) and n−m "failures"(event A) is a sum of elementary events:
B = A∩A∩ . . .∩A∩ A∩ A∩ . . .∩ A = AmAn−m
Due to independence of these events, the probability of one such series is:
P(B) = P(A) ·P(A) · . . . ·P(A) ·P(A) ·P(A) · . . . ·P(A) = pmqn−m
The event B is equally likely to the series of n trials containing m "successes" in di�erent order.There are as many such series as many possibilities to choose �positions� for "successes", that isin
(nm
)ways. According to the sum rule of independent events we obtain the Bernoulli's formula:
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 8 / 18
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Bernoulli's formula (also called BinomialProbability Formula)
Theorem
The probability Pm,n of achieving exactly m "successes" in n trials is
Pm,n =
(n
m
)pmqn−m =
n!
m!(n−m)!pmqn−m
where:
p is a probability of "success" in one trial;
q = 1−p is a probability of "failure" in one trial.
Example: We toss a coin 10 times. What is the probability we get 4 heads?
Here p = q = 0,5 and according to the Bernoulli's formula
P4,10 =10!
4!6!
(1
2
)4(1
2
)6
≈ 0,205
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 9 / 18
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Bernoulli's formula; example 2
We toss a coin 12 times. What is the probability we get 0 to 3 heads?
P = P0,12 +P1,12 +P2,12 +P3,12 =
=12!
0!12!
(1
2
)0(1
2
)12
+12!
1!11!
(1
2
)1(1
2
)11
+12!
2!10!
(1
2
)2(1
2
)10
+12!
3!9!
(1
2
)3(1
2
)9
=
=
(1
2
)12 [1+12+
12 ·112
+12 ·11 ·10
6
]=
=299
4096≈ 0,073
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 10 / 18
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Bernoulli's formula; example 3
Students are taking a 40 question multiple choice test. Each question has four choices. Astudent who had no time to prepare is going to guess all answers at random. What ismore likely: he will answer all questions correctly or he will give no right answers?
The probability that all choices are correct:
P40,40 =
(40
40
)·(1
4
)40
= 0,2540
and the probabilit that all answers are wrong:
P0,40 =
(40
0
)·(3
4
)40
= 0,7540.
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 11 / 18
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Binomial Distribution B(n,p)
Probability mass function:
f (x) =
(n
x
)pxqn−x =
(n
x
)px
(1−p)n−x,
where x = 0,1,2, . . . ,n.Mean: µ = np.
Variance: σ2 = np(1−p).
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 12 / 18
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Binomial Distribution B(n,p)
Probability mass function:
f (x) =
(n
x
)pxqn−x =
(n
x
)px
(1−p)n−x,
where x = 0,1,2, . . . ,n.Mean: µ = np.
Variance: σ2 = np(1−p).
Normal approximation:
N (np,np(1−p))
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 12 / 18
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Next subsection
1 Probability density, distribution
2 Probability distributions
Binomial distribution
Multinomial distribution
Normal distribution
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 13 / 18
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Multinomial Distribution
The multinomial distribution is a generalization of the binomial distribution.
For n independent trials each of which leads to a success for exactly one of k categories,with each category having a given �xed success probability, the multinomial distributiongives the probability of any particular combination of numbers of successes for the variouscategories.
When n = 1 and k = 2 the multinomial distribution is the Bernoulli distribution.
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 14 / 18
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Multinomial Distribution
The multinomial distribution is a generalization of the binomial distribution.
For n independent trials each of which leads to a success for exactly one of k categories,with each category having a given �xed success probability, the multinomial distributiongives the probability of any particular combination of numbers of successes for the variouscategories.
When n = 1 and k = 2 the multinomial distribution is the Bernoulli distribution.
Example
For example, suppose that two chess players had played numerous games and it was determinedthat the probability that Player A would win is 0.40, the probability that Player B would win is0.35, and the probability that the game would end in a draw is 0.25. The multinomialdistribution can be used to answer questions such as: �If these two chess players played 12games, what is the probability that Player A would win 7 games, Player B would win 2 games,and the remaining 3 games would be drawn?�
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 14 / 18
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Multinomial Distribution
The multinomial distribution is a generalization of the binomial distribution.
For n independent trials each of which leads to a success for exactly one of k categories,with each category having a given �xed success probability, the multinomial distributiongives the probability of any particular combination of numbers of successes for the variouscategories.
When n = 1 and k = 2 the multinomial distribution is the Bernoulli distribution.
Example
For example, suppose that two chess players had played numerous games and it was determinedthat the probability that Player A would win is 0.40, the probability that Player B would win is0.35, and the probability that the game would end in a draw is 0.25. The multinomialdistribution can be used to answer questions such as: �If these two chess players played 12games, what is the probability that Player A would win 7 games, Player B would win 2 games,and the remaining 3 games would be drawn?�
Probability for this case:
p =n!
(n1!)(n2!)(n3!)pn11pn22pn33
where n is the total number of eventsni is the number of times Outcome i occurs,p1 is the probability of Outcome i .
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 14 / 18
ioc.pdf
Multinomial Distribution
The multinomial distribution is a generalization of the binomial distribution.
For n independent trials each of which leads to a success for exactly one of k categories,with each category having a given �xed success probability, the multinomial distributiongives the probability of any particular combination of numbers of successes for the variouscategories.
When n = 1 and k = 2 the multinomial distribution is the Bernoulli distribution.
Example
For example, suppose that two chess players had played numerous games and it was determinedthat the probability that Player A would win is 0.40, the probability that Player B would win is0.35, and the probability that the game would end in a draw is 0.25. The multinomialdistribution can be used to answer questions such as: �If these two chess players played 12games, what is the probability that Player A would win 7 games, Player B would win 2 games,and the remaining 3 games would be drawn?�
For this example:
p =12!
(7!)(2!)(3!)0.470.3520.253 = 0.0248
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 14 / 18
ioc.pdf
Multinomial Distribution
The multinomial distribution is a generalization of the binomial distribution.
For n independent trials each of which leads to a success for exactly one of k categories,with each category having a given �xed success probability, the multinomial distributiongives the probability of any particular combination of numbers of successes for the variouscategories.
When n = 1 and k = 2 the multinomial distribution is the Bernoulli distribution.
Example
For example, suppose that two chess players had played numerous games and it was determinedthat the probability that Player A would win is 0.40, the probability that Player B would win is0.35, and the probability that the game would end in a draw is 0.25. The multinomialdistribution can be used to answer questions such as: �If these two chess players played 12games, what is the probability that Player A would win 7 games, Player B would win 2 games,and the remaining 3 games would be drawn?�
Formula for k outcomes:
p =n!
(n1!)(n2!) · · ·(nk !)pn11pn22· · ·pnkk
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 14 / 18
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Next subsection
1 Probability density, distribution
2 Probability distributions
Binomial distribution
Multinomial distribution
Normal distribution
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 15 / 18
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Introduction to Normal DistributionsThe normal distribution is the most important and most widely used distribution. It is alsocalled the �bell curve,� or �Gaussian curve�.
The density of the normal distribution is given by the function
f (x) =1√2πσ2
e−(x−µ)2
2σ2
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 16 / 18
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Features of normal distributions
1 Normal distributions are symmetric around their mean.
2 The mean, median, and mode of a normal distribution are equal.
3 The area under the normal curve is equal to 1.0.
4 Normal distributions are denser in the center and less dense in the tails.
5 Normal distributions are de�ned by two parameters, the mean (µ) and the standarddeviation (σ).
6 68% of the area of a normal distribution is within one standard deviation of the mean.
7 Approximately 95% of the area of a normal distribution is within two standard deviationsof the mean
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 17 / 18
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Central Limit Theorem
The normal distribution is important because of the following
Theorem (Central Limit Theorem)
Given a population with a �nite mean µ and a �nite non- zero variance σ2, the sampling
distribution of the mean approaches a normal distribution with a mean of µ and a variance of
σ2/N as N, the sample size, increases.
Most statistical procedures for testing di�erences between means assume extremelyimportant central limit theorem, the topic of a later section of this normal distributions.Because the distribution of means is very close to normal,
Quételet was the �rst to apply the normal distribution to human characteristics. He notedthat characteristics such as height, weight, and strength were normally distributed.
margarita.spitsakova@ttu.ee ICY0006: Lecture 6 18 / 18
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