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A game consists of tossing a fair coin 3 times. Let random variable W denote the number of heads obtained. We also let random variable Z denote the winnings earned in a single play of a game with the following rules, the outcomes of the 3 spins of the coin. • player wins $1 if first Head occurs on the first toss • player wins $2 if first Head occurs on the second toss • player wins $3 if first Head occurs on the third toss • player loses $1 if no Head occurs after 3 tosses. (a) What is the range of W (b) What is the range of Z (c) What is the joint PMF of pwz(w, z) (d) Are W and Z independent?

A game consists of tossing a fair coin 3 times. Let random variable W denote the number of heads obtained. We also let random variable Z denote the winnings earned in a single play of a game with the following rules, the outcomes of the 3 spins of the coin. • player wins $1 if first Head occurs on the first toss • player wins $2 if first Head occurs on the second toss • player wins $3 if first Head occurs on the third toss • player loses $1 if no Head occurs after 3 tosses. (a) What is the range of W (b) What is the range of Z (c) What is the joint PMF of pwz(w, z) (d) Are W and Z independent?

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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A game consists of tossing a fair coin 3 times. Let random variable \( W \) denote the number of heads obtained. We also let random variable \( Z \) denote the winnings earned in a single play of a game with the following rules, the outcomes of the 3 spins of the coin. - The player wins $1 if the first Head occurs on the first toss. - The player wins $2 if the first Head occurs on the second toss. - The player wins $3 if the first Head occurs on the third toss. - The player loses $1 if no Head occurs after 3 tosses. Questions: (a) What is the range of \( W \)? (b) What is the range of \( Z \)? (c) What is the joint PMF of \( p_{WZ}(w, z) \)? (d) Are \( W \) and \( Z \) independent?
Transcribed Image Text:A game consists of tossing a fair coin 3 times. Let random variable \( W \) denote the number of heads obtained. We also let random variable \( Z \) denote the winnings earned in a single play of a game with the following rules, the outcomes of the 3 spins of the coin. - The player wins $1 if the first Head occurs on the first toss. - The player wins $2 if the first Head occurs on the second toss. - The player wins $3 if the first Head occurs on the third toss. - The player loses $1 if no Head occurs after 3 tosses. Questions: (a) What is the range of \( W \)? (b) What is the range of \( Z \)? (c) What is the joint PMF of \( p_{WZ}(w, z) \)? (d) Are \( W \) and \( Z \) independent?
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