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AP Stats Notes
7.2 Day 1: Mean & Variance of Random Variables
Name_______________________________
Date__________________________
The Mean of a Random Variable
 The mean x of a set of observations is their ordinary average, but how do you
find the mean of a discrete random variable whose outcomes are not equally
likely?
Ex 1: The Tri-State Pick 3
 In the Tri-State Pick 3 game that New Hampshire shares with Maine and
Vermont, you choose a 3-digit number and the state chooses a 3-digit winning
number at random and pays you $500 if your number is chosen.
The probability distribution of X (the amount your ticket pays you)
Payoff X:
Probability:
$0
0.999
$500
0.001
 The ordinary average of the two possible outcomes is $250, but that makes no
sense as the average because $0 is far more likely than $500.
So what is the mean?


Mean of a Discrete Random Variable
 Suppose that X is a discrete random variable whose distribution is
Value of X: x1 x2 x3 … xk
Probability: ρ1 ρ2 ρ3 … ρk
 To find the mean of X, multiply each possible vlaue by its probability, then
add all the products
μx = x1ρ1 + x2ρ2 + … + xkρk
= Σxiρi
Ex 2: Benford’s Law
 What is the expected value of the first digit if each digit is equally likely?
First Digit X 1
2
3
4
5
6
7
8
9
Probability
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1
 What is the expected value if the data obeys Benford’s Law?
First Digit X 1
Probability
2
3
.301 .176 .125
4
5
6
7
8
9
.097
.079
.067
.058
.051
.046
Probability Histogram for equally likely outcomes 1 to 9
Probability Histogram for Benford’s Law
Recall …
 Computing a measure of spread is an important part of describing a
distribution (SOCS)
 The ________________ and the _______________________ are the measures
of spread that accompany the choice of the ___________ to measure center.
Variance of a Discrete Random Variable
 Suppose that X is a discrete random variable whose distribution is
Value of X: x1 x2 x3 … xk
Probability: ρ1 ρ2 ρ3 … ρk
2
 And that the mean μ is the mean of X. The variance of X is
 The standard deviation σx of X is the square root of the variance.
Ex 3: Linda Sells Cars
 Linda is a sales associate at a large auto dealership. She motivates herself by
using probability estimates of her sales. For a sunny Saturday in April, she
estimates her car sales as follows:
Cars Sold: 0
1
2
3
Probability: 0.3
0.4
0.2
0.1
Find the mean and variance.
xi
0
1
2
3
pi
0.3
0.4
0.2
0.1
xipi
0.0
0.4
0.4
0.3
(xi – μx)2pi
The Law of Large Numbers
 Draw independent observations at random from any population with finite
mean μ.
 Decide how accurately you would like to estimate μ.
 As the number of observations drawn increases, the mean x of the observed
values eventually approaches the mean μ of the population as closely as you
specified and then stays that close.
Ex 4: Heights of Young Women (Law of Large Numbers)
3
How large is a large number?
 The law of large numbers does not state how many trials are necessary to
obtain a mean outcome that is close to μ.
 The number of trials depends on the _________________________________.
 The more variable the outcomes, ____________________________________
_______________________________________________________________
_______________________________________________________________.
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