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Chapter 1: Problem 3

Cast a die a number of independent times until a six appears on the up side of the die. (a) Find the pmf \(p(x)\) of \(X\), the number of casts needed to obtain that first six. (b) Show that \(\sum_{x=1}^{\infty} p(x)=1\). (c) Determine \(P(X=1,3,5,7, \ldots)\). (d) Find the cdf \(F(x)=P(X \leq x)\).

Short Answer

Expert verified
(a) \( p(x) = (5/6)^(x - 1) * (1/6) \) (b) The sum of all probabilities equals to one in geometric distribution. (c) The sum of probabilities for all odd numbers can be calculated as a series. (d) \( F(x) = 1 - (5/6)^x \)

Step by step solution

01

Find the pmf

The situation can be modeled with a geometric distribution, where the pmf is given by \( p(x) = (1 - p)^(x - 1) * p \), where \( p \) is the probability of success, which is 1/6 for a die. So, \( p(x) = (5/6)^(x - 1) * (1/6) \)
02

Show the sum of all probabilities equals to one

The sum of all probabilities for a pmf should equal to one. For a geometric distribution, this is valid, since \(\sum_{x=1}^{\infty} (1 - p)^(x - 1) * p = 1\).
03

Determine \(P(X=1,3,5,7, \ldots)\)

This is the sum of the probabilities for all odd numbers. As each throw is independent, the probability \( p(x) = (5/6)^(x - 1) * (1/6) \) applies to each odd number and the sum of these probabilities can be calculated as a series.
04

Find the cdf

The cumulative distribution function (CDF) gives the probability that the variable is equal or less than a certain value. For a geometric distribution it is given by \( F(x) = 1 - (1 - p)^x \), so in this case, \( F(x) = 1 - (5/6)^x \).

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