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I just need b, c, and d thanks!
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I just need b, c, and d thanks!
I just need b, c, and d thanks!
5. Consider a gambler who starts with an initial fortune of $1 and then on each successive gamble, independent of the past outcomes, either wins $1 with probability p or loses $1 with probability q = 1 - p. Let X, denote their total fortune after the gamble. The gambler's objective is to reach a total fortune of SN, without first getting ruined (running out of money). If the gambler succeeds, then the gambler is said to win the game. In any case, the gambler stops playing after winning or getting ruined, whichever happens first. There is nothing special about the gambler starting with $1. and so we consider the more general case where the gambler starts with Sd where 0 < d <N. (a) To make things more trackable, we define Xo = d and break up Xn = A + A2 + ... + A, where the random variables (A1,...,A} are the independent outcomes of each of the n identical gambles. Give the probability mass function of the random variable A which gives the payoff amount for a single gamble. 1 for x=P Palla 1 for x 2 o othonrise (b) What is the expected value of A? What does it mean in the context of the game if this value positive, negative, or zero? Elx) = le more the last -lg=2P-1 if verive is online, in mars he word die negative it has means lost chore it he he last So the same amount as (c) What is the variance and standard deviation of A? (d) Since the game eventually stops when either the gambler goes bust (Xn = 0) or reaches their goal fortune ( X = N), we define a random variable to track how long the game lasts T = min {n > 1 such that X, € {0, N}} What is the probability that the gambler reaches their goal fortune? That is, XT = N?
5. Consider a gambler who starts with an initial fortune of $1 and then on each successive gamble, independent of the past outcomes, either wins $1 with probability p or loses $1 with probability q = 1 - p. Let X, denote their total fortune after the gamble. The gambler's objective is to reach a total fortune of SN, without first getting ruined (running out of money). If the gambler succeeds, then the gambler is said to win the game. In any case, the gambler stops playing after winning or getting ruined, whichever happens first. There is nothing special about the gambler starting with $1. and so we consider the more general case where the gambler starts with Sd where 0 < d <N. (a) To make things more trackable, we define Xo = d and break up Xn = A + A2 + ... + A, where the random variables (A1,...,A} are the independent outcomes of each of the n identical gambles. Give the probability mass function of the random variable A which gives the payoff amount for a single gamble. 1 for x=P Palla 1 for x 2 o othonrise (b) What is the expected value of A? What does it mean in the context of the game if this value positive, negative, or zero? Elx) = le more the last -lg=2P-1 if verive is online, in mars he word die negative it has means lost chore it he he last So the same amount as (c) What is the variance and standard deviation of A? (d) Since the game eventually stops when either the gambler goes bust (Xn = 0) or reaches their goal fortune ( X = N), we define a random variable to track how long the game lasts T = min {n > 1 such that X, € {0, N}} What is the probability that the gambler reaches their goal fortune? That is, XT = N?
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