# Section 3 3 3 5 Expected Value Binomial and Hypergeometric Random Variables Example 2 Poker hands Hand Obtained Probability Payoff Let X be the payoff Nothing 0 50 by hmb10625

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Section 3.3-3.5

Expected Value, Binomial, and
Hypergeometric Random Variables
Example 2. Poker hands.
Hand Obtained Probability Payoff   Let X be the payoff
Nothing         0.501177       0   earned with one poker
One Pair        0.422569     0.5   hand. What is the pmf of
Two Pairs       0.047539       2   X?
Three of a Kind 0.021128       5
Straight        0.003925      10
If you play the game
Flush           0.001965      20
many times what average
Full House      0.001441      25
payoff would you
Four of a Kind   0.00024    100
expect?
Straight Flush  0.000014 1000
Royal Flush     0.000002 10000
In lab, compared to
payoff of 10, 100 games.
The Expected Value of a Function
Let X be a discrete random variable with p.m.f. p(x). The
expected value or mean of X is given by

E( X )   X      x  p ( x)
x

•If X is the payoff of a poker hand, what is E(X)?

•If X is the sum of two six sided die, what is E(X)?
Example 3.
•Let X be the sum of two fair 6-sided dice. What are the
possible values of X2? What is the probability of each of these
outcomes?

•Calculate E(X2).

•Expected value of a function. Let X be a discrete r.v. with
p.m.f. p(x) and let h(x) be a function. Then

E (h( X ))     h( x)  p( x)
x
Mean, Variance and Standard Deviation
• Let X be a discrete random variable.
• We define the mean of X to be E[X] (usually denoted ).
• We define the variance of X to be
2 = E[(X  )2] = …=E[X2]  E[X]2.
(The first equality is taken as the definition of variance.
The last formula is used for computations.)
• We define the standard deviation of X to be  = 2 .
• These are theoretical versions of the sample mean, sample
variance and sample standard deviation introduced for
data.
Example 3. Continued.

Let X be the sum of two fair six sided die. Recall E(X). and
E(X2).

Calculate the variance from the definition.

Calculate the variance from the shortcut formula.

What is the standard deviation?
In Minitab, generate 100 realizations of X. Calculate the mean
and standard deviation of the data.
Summary of 3.3

• Use the p.m.f. to calculate expected value of
a random variable.
• Calculate expected value of a function of a
random variable.
• Mean, Variance, and Standard Deviation of
a random variable (, 2, ).
• Mean, variance, standard deviation of a data
set (x_bar, s2, s)
Binomial Random Variables
• Recall what binomial models.
• Think of an application of the binomial r.v.

• What is the pmf of a binomial r.v.? Note
instead of p(x), pmf is labeled b(x;n,p).
CDF of Binomial R.V.
• Instead of F(x), cdf is labeled B(x;n,p). The
cdf is tabulated in A.1. Part given here.
• Suppose X is Bin(n,p),
n=10
–   find P(X2) from pmf.   x           p=.01     p=.05     p=.1
–   find P(X2) from cdf.           0
1
0.904
0.996
0.599
0.914
0.349
0.736
–   Find P(2 X 5) from pmf.       2       1.000     0.988      0.930
3       1.000     0.999      0.987
–   Find P(2 X 5) from cdf.       4       1.000     1.000      0.998
5       1.000     1.000      1.000
6       1.000     1.000      1.000
7       1.000     1.000      1.000
8       1.000     1.000      1.000
9       1.000     1.000      1.000
10       1.000     1.000      1.000
Mean and Variance of Binomial
• Give summations necessary for finding
mean and variance of a random variable.

• Summations simplify to
– The mean of a binomial is np.
– The variance is np(1-p).
Hypergeometric Random Variables
• Recall what hypergeometric probabilities are used
for. Think of an application of hypergeometric
probabilities.

• Let X be the number of successes obtained when
sampling from a population of size N with M
successes and N-M failures. X is a hypergeometric
r.v. What is the pmf of X? Note instead of p(x),
pmf is labeled h(x;n,M,N).
Mean and Variance
• The mean of a hypergeometric r.v. is
M
n
N
• The variance is np(1-p).

 N n M  M 
       n 1  
 N 1  N    N
Relationship between binomial
and hypergeometric.
• One application of binomial is sampling
from finite population of S and F with
replacement.
• While the hypergeometric is sampling w/o
replacement.
Recall HW problem:
• 10000 boards, 2000 of which are green.
Sample 2 boards . Are the events “1st
board is green” and “second board is green”
independent if sampling is done
– w/ replacement
– w/o replacement
– Suppose instead there are only 10 boards, 2 of
which are green and the sampling is done
w/replacement.
– W/o replacement
Compare mean and variance.
• Set p=M/N. Sample size of n.
• What is the expected number of successes if
sampling is done with replacement?
Without replacement?

• What is the variance of the number of
successes if sampling is done with
replacement? Without replacement?
Notes
• You should know and/or be able to derive pmfs of
Binomial and Hypergeometric random variables.
• You should know mean and variance of each.
• You should be able to use tabulated cdf for
Binomial random variables.
• You are not responsible for the negative binomial
r.v. described in section 3.5 or the Poisson r.v
described in section 3.6. However, you should be
able to work with these and potentially other
random variables if details are given.

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