Question

# Suppose that there are 5 dollar bills in a box: three 1 dollar bills, one 5...

Suppose that there are 5 dollar bills in a box: three 1 dollar bills, one 5 dollar bill and one 10 dollar bill. You are allowed to pick up two bills at the same time from the box randomly. Let X denote the money you get from this game. (a) What’s the support and p.m.f. of X? (b) What’s the mean and variance of X?

Let X is a random variable shows the amount you win excluding you pay to play. Here X can take values \$2, \$6 and \$11,\$15.

Total number of bills: 3+1+1=5

Number of ways of selecting 2 bills out of 5 is C(5,2) = 10

When X=\$2, then both \$1 bills are selected.

So

P(X=\$2) = C(3,2) / 10 = 3/10 =0.30

When X=\$6, then one \$5 bill and one \$1 bill is selected.

So

P(X=\$6) = [C(3,1)*C(1,1)] / 10 = 0.30

When X=\$11, then one \$10 bill and one \$1 bill is selected.

So

P(X=\$10) = [C(3,1)*C(1,1)] / 10 = 0.30

When X=\$15, then one \$5 bill and one \$10 bill is selected.

So

P(X=\$15) = [C(1,1)*C(1,1)] / 10 = 0.10

The pmf of X is:

 X P(X=x) 2 0.3 6 0.3 11 0.3 15 0.1

(b)

Following table shows the calculations for mean and variance:

 X P(X=x) xP(X=x) x^2P(X=x) 2 0.3 0.6 1.2 6 0.3 1.8 10.8 11 0.3 3.3 36.3 15 0.1 1.5 22.5 Total 7.2 70.8

The mean is:

The variance is:

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