CK-12 Probability and Statistics - Advanced

http://www.ck12.org Chapter 7. Sampling Distributions and Estimations

μX+Y+Z=μX+μY+μZ

μX+Y+Z=$2. 60 +$2. 60 +%2. 60

μX+Y+Z=$7. 80

The expected value is the same as that for the tripled prize.

Since the winnings on the three games played are independent, the standard deviation ofX+Y+Zis:

σ^2 X+Y+Z=σ^2 X+σ^2 Y+σ^2 Z

σ^2 X+Y+Z= 6. 462 + 6. 462 + 6. 462

σ^2 X+Y+Z≈ 125 .1948 and σ≈

√

125. 1948 ≈ 11. 19

The person playing the three games expects to win $7.80 with a standard deviation of $11.19. When the prize was
tripled, there was a greater standard deviation($19. 36 )than when the person played three games($11. 19 ).

The rules for addition and subtraction for random variables are:

IfXandYare random variables then:

μX+Y=μX+μY

μX−Y=μX−μY

IfXandYare independent then:

σ^2 X+Y=σ^2 X+σ^2 Y

σ^2 X−Y=σ^2 X+σ^2 Y

Variances are added for both the sumanddifference of two independent random variables because the variation in
each variable contributes to the variation in each case. Subtracting is the same as adding the opposite. Suppose
you have two dice, one die(X)with the normal positive numbers 1 through 6, and another(Y)with the negative
numbers−1 through−6. Then suppose you perform two experiments. In the first, you roll the first die(X)and then
the second die(Y), and you compute the difference of the two rolls. In the second experiment you roll the first die
(X)and then the second die(Y)and you calculate the sum of the two rolls.