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7.2 Means and Variances of Random Variables
Objective: Calculate the mean and standard deviation of random variables
Understand the law of large numbers
The mean of a Random Variable
Ex:
You pick a three digit number in the lottery. If your number matches the states number, you
win $500. What are your average winnings?
Probability Distribution
Outcome
Probability
Probabilities are an idealized description of long-run proportions, so the mean of a probability
distribution describes the____________________________________.
Mean of a Discrete Random Variable
Suppose X is a discrete random variable whose distribution is
Value of X
Probabiltiy
x₁
p₁
x₂
p₂
x₃
p₃
....
....
𝑥𝑘
𝑝𝑘
To find the mean of X, multiply each possible value by its probability, then add all the products:
𝜇𝑥 = 𝑥1 𝑝1 + 𝑥2 𝑝2 + 𝑥3 𝑝3 + ⋯ . +𝑥𝑘 𝑝𝑘
Ex: The distribution of the count X of heads in four tosses of a balanced coin.
Numbers of
Heads
Probability
0
1
2
3
4
0.0625
0.25
0.375
0.25
0.0625
The expected value is:
𝜇𝑥 =
Construct a histogram of the distribution.
What do you notice about the mean?
Ex: What is the mean number of inhabitants in an American household?
Inhabitants 1
Proportion 0.25
of
households
2
0.32
3
0.17
4
0.15
5
0.07
6
0.03
7
0.01
Complete 7.17-7.19 (pg.389)
7.17
7.18
7.19
Statistical estimation and the law of large numbers
Law of large numbers
Draw independent observations at random from any population with finite mean (μ).
Decide how accurately you would like to estimate the mean. As the number of observations
drawn increases, the mean of the observed values eventually approaches the mean of the
population as closely as you specified and then stays that close.
Describe this in your own words?
Reece’s example: http://www.rossmanchance.com/applets/Reeses3/ReesesPieces.html
Ex: Use the average height of women to explain this. The mean is 64.5 in with a standard
deviation of 2.5 in.
Rules for Means
Rule 1: If X is a random variable and a and b are fixed numbers, then
𝝁𝒂+𝒃𝒙 =
Rule 2: If X and Y are (independent) random variables, then
𝝁𝒙+𝒚 =
Ex: Military divisions
Units Sold
Probability
1000
0.1
3000
0.3
5000
0.4
10000
0.2
Civilian Division
Units Sold
Probability
300
0.4
500
0.5
750
0.1
Let x= # of military units sold
y= # of civilian units sold
What is the mean number of military units sold? Civilian units sold?
If a profit of $2000 is made on each military unit sold and $3500 is made on each civilian unit,
what is the total mean profit for units sold?
Variance of a Discrete Random Variable
The variance of X is:
Value of X
Probabiltiy
x₁
p₁
x₂
p₂
x₃
p₃
....
....
𝑥𝑘
𝑝𝑘
𝜎𝑥2 = (𝑥1 − 𝜇𝑥 )2 𝑝1 + (𝑥2 − 𝜇2 )2 𝑝2 +(𝑥3 − 𝜇3 )2 𝑝3 + ⋯ + (𝑥𝑘 − 𝜇𝑘 )2 𝑝𝑘
So the standard deviation is:
√𝜎2𝑥
Steps in the calculator:
L₁
L₂
L₃
x
p
(𝐿1 − 𝜇𝑥 )2 𝐿2
Then sum(L₃)
Find the standard deviation of the military units sold?
Find the standard deviation of the civilian units sold?
Rules for Variance
Rule 1:
𝝈𝟐𝒂+𝒃𝒙 =
Rule 2:
𝝈𝟐𝒙+𝒚 =
***___________________________________________________________________***
Ex: The payoff X of a $1 ticket in the Tri-State pick 3 game is $500 with probability 1/1000 and
$0 the rest of the time. What is the variance of the total payoff if you buy $1 ticket on two
different days?
SAT scores
SAT math score X
SAT verbal score Y
𝜇𝑥 = 625 𝜎𝑥 = 90
𝜇𝑥 = 590 𝜎𝑥 = 100
What are the mean and standard deviation of the total score X + Y among students applying to
this college?
Golf scores
Tom’s score X: 𝜇𝑥 = 110 𝜎𝑥 = 10
George’s score Y: 𝜇𝑥 = 100 𝜎𝑥 = 8
Their scores vary independently. What is the mean difference between their scores?
What is the variance of the difference between their scores?
So the standard deviation is?
The class average on the last chapter test was 80 with a standard deviation of 4.
What is the mean if I doubled everyone’s test score?
What is the standard deviation?
What if I added 5 bonus points to everyone’s score. What is the new mean and standard
deviation?
What if I doubled everyone’s score and added 5 points. What is the new mean and standard
deviation?