1. Probability Distributions
Chapter 4
Probability Distributions
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2. Discrete Random Variables
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Random Variable
For a given sample space S of some experiment, a random variable is any rule that associates a number with each outcome in S .
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Types of Random Variables
A discrete random variable is an rv whose possible values either constitute a finite set or else can listed in an infinite sequence. A random variable is continuous if its set of possible values consists of an entire interval on a number line.
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Random Variables
Represents a possible numerical value from a random event
Random
Variables
Discrete
Random Variable
Continuous
Random Variable
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6. Probability Distributions for Discrete Random Variables
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7. Discrete Random Variables
Can only assume a countable number of values
Examples:
Roll a die twice
Let x be the number of times 4 comes up
(then x could be 0, 1, or 2 times)
Toss a coin 5 times.
Let x be the number of heads
(then x = 0, 1, 2, 3, 4, or 5)
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8. Discrete Probability Distribution
Experiment: Toss 2 Coins. Let x = # heads.
4 possible outcomes
Probability Distribution T T
x Value Probability
/4 = .25
/4 = .50
/4 = .25 T H H T
.50
.25
Probability H H x
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9. Discrete Probability Distribution
The set of ordered Pairs (x, f(x) ) is a
probability function, probability mass function or probability distribution function of the discrete random variable X if
f(x) 0
P(X=x) = f(x)
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Cumulative Distribution Function
The cumulative distribution function (cdf) F(x) of a discrete rv variable X with pmf p(x) is defined for every number by
For any number x, F(x) is the probability that the observed value of X will be at most x.
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Proposition
For any two numbers a and b with
“a–” represents the largest possible X value that is strictly less than a.
Note: For integers
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Example
A probability distribution for a random variable X: x –8 –3 –1 1 4 6
P(X = x)
0.13
0.15
0.17
0.20
0.11
0.09
Find
0.65
0.67
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13. Expected Values of Discrete Random Variables
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The Expected Value of X
Let X be a discrete rv with possible values x and pmf p(x). The expected value or mean value of X, denoted
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Ex. Use the data below to find out the expected number of the number of credit cards that a student will possess.
x = # credit cards x
P(x =X)
0.08 1
0.28 2
0.38 3
0.16 4
0.06 5
0.03 6
0.01
=1.97
About 2 credit cards
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The Expected Value of a Function
If the rv X has the set of possible values x and pmf p(x), then the expected value of any function h(x), denoted
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The Variance and Standard Deviation
Let X have pmf p(x), and expected value Then the variance of X, denoted V(X)
The standard deviation (SD) of X is
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Ex. The quiz scores for a particular student are given below:
22, 25, 20, 18, 12, 20, 24, 20, 20, 25, 24, 25, 18
Find the variance and standard deviation.
Value 12 18 20 22 24 25
Frequency 1 2 4 3
Probability
.08
.15
.31
.23
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Shortcut Formula for Variance
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Mean and Variance of Linear Combinations of Random Variables
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Expected Value of Linear Combinations
This leads to the following:
For any constant a,
2. For any constant b,
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The Expected Value of
Also
Finally
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If X and Y are independent, then
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Rules of Variance
and
This leads to the following:
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If X and Y are random variables, then
If X and Y are Independent random variables, then
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If X and Y are Independent random variables, then
If X1, X2, … Xn are Independent random variables, then
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28. Special Distributions for Discrete Random Variables
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29. The Discrete Uniform Distribution
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If the random variable X assumes the values x1, x2, …, xk with equal probability, then the discrete uniform distribution is given by
The mean and the variance of the discrete uniform are respectively
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31. The Binomial Probability Distribution
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Binomial Experiment
An experiment for which the following four conditions are satisfied is called a binomial experiment (Bernoulli Process).
The experiment consists of a sequence of n trials, where n is fixed in advance of the experiment.
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The trials are identical, and each trial can result in one of the same two possible outcomes, which are denoted by success (S) or failure (F).
The trials are independent.
The probability of success is constant from trial to trial: denoted by p.
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Binomial Random Variable
Given a binomial experiment consisting of n trials, the binomial random variable X associated with this experiment is defined as
X = the number of S’s among n trials
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35. Notation for the pmf of a Binomial rv
Because the pmf of a binomial rv X depends on the two parameters n and p, we denote the pmf by b(x;n,p).
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36. Computation of a Binomial pmf
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Ex. A card is drawn from a standard 52-card deck. If drawing a club is considered a success, find the probability of
a. exactly one success in 4 draws (with replacement).
p = ¼; q = 1– ¼ = ¾
b. no successes in 5 draws (with replacement).
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Notation for cdf
For X ~ Bin(n, p), the cdf will be denoted by
x = 0, 1, 2, …n
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Mean and Variance
For X ~ Bin(n, p), then
E(X) = np, and
V(X) = np(1 – p) = npq,
(where q = 1 – p).
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40. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. 5 cards are drawn, with replacement, from a standard 52-card deck. If drawing a club is considered a success, find the mean, variance, and standard deviation of X (where X is the number of successes).
p = ¼; q = 1– ¼ = ¾
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41. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. If the probability of a student successfully passing this course (C or better) is 0.82, find the probability that given 8 students
a. all 8 pass.
b. none pass.
c. at least 6 pass. =
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42. Hypergeometric and Distributions
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The Hypergeometric Distribution
The three assumptions that lead to a hypergeometric distribution:
1. The population or set to be sampled consists of N individuals, objects, or elements (a finite population).
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Each individual can be characterized as a success (S) or failure (F), and there are K successes in the population.
A sample of n individuals is selected without replacement in such a way that each subset of size n is equally likely to be chosen.
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Hypergeometric Distribution
If X is the number of S’s in a completely random sample of size n drawn from a population consisting of K S’s and (N – K) F’s, then the probability distribution of X, called the hypergeometric distribution, is given by
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46. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. A card is drawn from a standard 52-card deck. If drawing a club is considered a success, find the probability of
a. exactly one success in 4 draws (without replacement).
N=52, K=13, n=4
b. no successes in 5 draws (without replacement).
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47. Hypergeometric Mean and Variance
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Geometric Distribution
The pmf of the geometric rv X with parameter p = P(S) is
x = 1, 2, …
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49. Geometric Mean and Variance
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50. The Poisson Probability Distribution
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Poisson Process
Assumptions:
1. The number of outcomes occurring in one time interval or specified region of space is independent of the number that occurs in any other disjoint time interval or specified region.
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Poisson Process
2. The probability that a single outcome will occur during a very short time interval or specified region is proportional to the length of time interval or the size of the region and does not depend on the number of outcomes occurring outside interval or specified region
3.The Probability that more than one outcome will occur in such short time interval or fall in such small region is negligible.
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53. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Poisson Distribution
The probability Distribution of the Poisson random variable X, representing the number of outcomes occurring in a given time interval or specified region denoted by t is
where is the average number of outcomes per unit time or region.
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54. The Poisson Distribution as a Limit of the binomial
Suppose that in the binomial pmf b(x;n, p), we let in such a way that np approaches a value
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55. Poisson Distribution Mean and Variance
If X has a Poisson distribution with parameter
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56. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. At a department store customers arrive at a checkout counter at a rate on three per 15 minutes. What is the probability that a counter has two customers over a 30 minutes randomly selected?
Sol.
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