AP Statistics
Mrs. Diaz
NAME:______________________________
DATE:______________________________
ASSIGNMENT #11
variances
Rules for means and
PART 1: Relationship between mean and variance
Suppose X has the distribution …
X
Prob.
2
0.25
3
0.75
And Supposed Y has the distribution…
Y
Prob.
4
0.30
5
0.20
6
0.50
Show that each of the following holds.
𝜎 2𝑋 = 0.187
𝜎 2 𝑌 = 0.760
𝜇𝑋 = 2.75
𝜇𝑌 = 5.20
Show that the distribution of 3X has a mean of 8.25 and variance of 1.687
3X
Prob.
6
0.25
9
0.75
Show that the distribution of 5Y has a mean of 26 and variance of 19.0
5Y
Prob.
20
0.30
25
0.20
30
0.50
𝜎𝑋 = 0.433
𝜎𝑌 = 0.872
Show that the distribution of 2 + 7X has a mean of 21.25 and variance of 9.187
2 + 7X
Prob.
16
0.25
23
0.75
In general, if X is a random variable then
𝜎 2 𝑎+𝑏𝑋 =
𝜇𝑎+𝑏𝑋 =

𝜎𝑎+𝑏𝑥 =
Do these rules for finding the mean and variance of 3 + 2Y? Try it.
PART 2: Adding mean and variances
Suppose X has the distribution …
X
Prob.
2
0.25
3
0.75
And Supposed Y has the distribution…
Y
Prob.
4
0.30
5
0.20
6
0.50
Show that each of the following holds.
𝜇𝑋 = 2.75
𝜇𝑌 = 5.20
𝜎 2𝑋 = 0.187
𝜎 2 𝑌 = 0.760
𝜎𝑋 = 0.433
𝜎𝑌 = 0.872
Let X + Y be the distribution of the sum of X and Y. Assume that X and Y are independent.
Show that the distribution of X + Y is:
X+Y
Prob.
6
0.075
7
0.275

Explain how the values in the table are determined.

Why is it important that X and Y be independent?

Find the mean and variance of X + Y.
𝜇𝑋+𝑌 =
𝜎 2𝑋+𝑌 =
8
0.275
9
0.375
𝜎𝑋+𝑌 =
In general, if X and Y are independent random variables, then
𝜇𝑋+𝑌 =

𝜎 2𝑋+𝑌 =
Do these hold true for subtraction?
𝜎𝑋+𝑌 =
PART 3: Subtracting means and variances
Again, we have shown that each of these following holds true.
𝜎 2𝑋 = 0.187
𝜎 2 𝑌 = 0.760
𝜇𝑋 = 2.75
𝜇𝑌 = 5.20
X-Y
Prob.
-1
0.225
-2
0.225
-3
0.425

Explain how the values in the table are determined.

Why is it important that X and Y be independent?
𝜎𝑋 = 0.433
𝜎𝑌 = 0.872
-4
0.125
Find the mean and variance of X – Y.
𝜇𝑋−𝑌 =
𝜎 2𝑋−𝑌 =
𝜎𝑋−𝑌 =
In general, if X and Y are independent random variables, then
𝜇𝑋−𝑌 =
𝜎 2𝑋−𝑌 =
𝜎𝑋−𝑌 =
GENERAL RULE: If X and Y are random variables and X and Y are
independent, then
𝝁𝑿±𝒀 =
𝝈𝟐 𝑿±𝒀 =
𝝈𝑿±𝒀 =
PART 4: PRACTICE OPEN RESPONSE
We have established rules for adding and subtracting discrete random variables. Do these same rules
apply for continuous random variables?
Suppose the heights of the boys and girls at a large high school are normally distributed with means
and standard deviations given in the table
Heights (inches)
Mean
Girls
Boys
64
68
Standard
Deviation
4
3
One boy and one girl are randomly selected. Which is more likely: to select a girl who is greater than
66 inches or a boy who is less than 66 inches tall? Explain.
One boy and one girl are randomly selected. What is the probability that the girl is taller than the boy?
One girl is selected. What is the probability she is taller than her brother?