Download §2.2 Probability Mass Functions

The probability mass function Hypergeometric random variables Bernoulli random variables Binomial random variables Geometric ran
§2.2 Probability Mass Functions
Tom Lewis
Fall Semester
2016
The probability mass function Hypergeometric random variables Bernoulli random variables Binomial random variables Geometric ran
Outline
The probability mass function
Hypergeometric random variables
Bernoulli random variables
Binomial random variables
Geometric random variables
The Poisson PMF
The probability mass function Hypergeometric random variables Bernoulli random variables Binomial random variables Geometric ran
Definition
Let X be a discrete random variable.
• To each x ∈ R, let
pX (x ) = P (X = x ).
pX is called a discrete density function or the probability mass
function, hereafter PMF.
• The set
{x ∈ R : pX (x ) > 0}
is called the support or the set of possible values of pX .
The probability mass function Hypergeometric random variables Bernoulli random variables Binomial random variables Geometric ran
Problem
An urn contains n = w + b balls: w white and b black. An
experiment consists of selecting k 6 n balls from the urn without
replacement. Let W count the number of white balls in this
sample. Show that the PMF for W is
 w b 
 j k−j
if j = 0, 1, . . . , k ;
n
pW (j ) =
k


0
if else.
W is an example of a hypergeometric random variable.
The probability mass function Hypergeometric random variables Bernoulli random variables Binomial random variables Geometric ran
Problem
An urn contains 10 balls numbered 1 through 10. An experiment
consists of selecting 2 balls from the urn without replacement. Let
m denote the smaller of the two numbers on the selected balls.
Find the PMF for m.
The probability mass function Hypergeometric random variables Bernoulli random variables Binomial random variables Geometric ran
Problem
An experiment consists of tossing a biased coin (chance p of
landing heads and chance q = 1 − p of landing tails). Let X = +1
if the coin lands heads and 0 if tails. Find the PMF for X .
X is an example of a Bernoulli random variable.
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Problem
Consider a Bernoulli process with n trials and success probability p.
Let
X = number of successes in the n trials
Show that
 n k n−k



 k p q
pX (k ) =




0
if k = 0, 1, 2, 3, . . . , n
if otherwise.
This is the binomial PMF with parameters n and p.
X is called a binomial random variable.
The probability mass function Hypergeometric random variables Bernoulli random variables Binomial random variables Geometric ran
Problem
A fair die is cast 10 times. Let X count the number of times a 1 or
a 6 was cast. Find the PMF for X .
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Problem
Consider a Bernoulli process with success probability p. Let this
process run until the first success is encountered. Let
X = number of trials until the first success
Show that
pX (k ) =
q k −1 p
if k = 1, 2, 3, . . .
0
if otherwise.
X is an example of a geometric random variable.
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Problem
Let X be a geometric random variable with parameter p.
• Show that P (X > m) = q m , m = 1, 2, 3, · · · .
• Given n, m > 1, show that
P (X > m + n | X > m) = P (X > n)
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Definition
Let > 0 and define
k
p(k ) =
k! e
0
−
if k = 0, 1, 2, . . .
if else.
p is a mass function called the Poisson PMF with parameter . Any
random variable with this PMF is called a Poisson random variable.
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Theorem (The Poisson approximation of the binomial)
Let Sn have a binomial PMF with parameters n and p. Let = np.
If n → ∞ and is fixed, then
k
P (Sn = k ) →
k!
e −.
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Problem (Poisson approximation)
It is well known that the probability that a light bulb is defective is
.02. Approximately what is the probability that a case of 75 light
bulbs will contain 3 defective bulbs?