On a certain game show you have already won 1000 The host in

On a certain game show, you have already won $1000. The host invites you to trade your SI000 for \"whatever\'s behind Door #1\". From past history, you know that Door #1 will have a prize worth $10,000 with probability 0.1, and one worth $500 withprobability 0.9. Let X = 1 if the grand prize appears ($10K),and 0 if the lesser prize turns up. Name the distribution of X, completely specify its pmf, and give its expected value and variance. Let W = your net winnings if you open the door. Express W as a linear function of X. What are your expected net winnings if you open the door? What is the standard deviation of your net winnings?

Solution

x =1 if prize is 10000

=0 if prize is for 500

P(X=1) = 0.1

P(X=0) = 0.9

Expected value of X = 1(0.1)+0(0.9)= 0.1

E(X^2) = 0.1

Variance =0.1-0.01 = 0.09

Std dev = 0.3

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W net winnings

= 10000 if x =1

= 500 if x =0

W = 10000 or 500

P(10000) = 0.1 and P(500) = 0.9

Expected winnings = 10000(0.1) = 500(0.9)

= 1000+450

=1450

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E(X^2) =10000000+225000

= 10225000

Var(W) = 10225000-1450^2

=8122500

Std dev =2850