Problem 203 3 points We have a nickel a dime and a quarter i

Problem 20.3 [3 points]. We have a nickel, a dime and a quarter in a cup. (a) We withdraw one coin. What is the expected amount of money we withdraw? What is the variance? Now we draw a second coin, without replacing the first. What is the expected amount of money we withdraw? What is the variance? (b) Suppose instead we consider withdrawing two coins from the cup together. What is the expected amount of money we withdraw, and what is the variance? (c) What does this example show about whether the variance of a sum of random variables is the sum of their variances.

Solution

one dime, one nickel and one quarter.

a)

Expected value = 40/3 = 13.33

E(X^2) =250

Var = 250-13.33^2 = 72.311

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b) Withdrawing 2 coins together

Then sample space (dime, nickel) (nickel, quarter) (dime, quarter)

Mean = 26.64

Var(x) = 782.55-26.64^2 = 72.8604

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Variance of sum of two variables is almost equal but sum is more than a single variable

	Dime	Nickel	Quarter	Total
Value	10	5	25
Prob	1/3	1/3`	1/3	1
value*pr	3 1/3	1 2/3	8 1/3	13 1/3