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Chapter 5: Q6E (page 302)

Suppose that X has the geometric distribution with parameter p. Determine the probability that the value ofX will be one of the even integers 0, 2, 4, . . . .

Short Answer

Expert verified

\[\sum\limits_{i = 0}^\infty {P\left[ {X = 2i} \right]} = \frac{1}{{2 - p}}\]

Step by step solution

01

Given information

Here, X is a random variable that follows a geometric distribution\(X \sim Geom\left( p \right)\).

02

Find the pdf of the distribution

\[P\left( {X = k} \right) = {\left( {1 - p} \right)^k} \times p,k = 0,1,2, \ldots \]

03

To find the probability that X takes even values.

\[\begin{array}{c}\sum\limits_{i = 0}^\infty {P\left[ {X = 2i} \right]} \\ = \sum\limits_{i = 0}^\infty {p{{\left( {1 - p} \right)}^{2i}}} \\ = p\sum\limits_{i = 0}^\infty {{{\left( {1 - p} \right)}^{2i}}} \\ = \frac{p}{{1 - {{\left( {1 - p} \right)}^2}}}\\ = \frac{1}{{2 - p}}\end{array}\]

Hence the answer is\[\frac{1}{{2 - p}}\]

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