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Chapter 4: Q4.2-10E (page 225)

Suppose that a fair coin is tossed repeatedly until a head is obtained for the first time.

(a) What is the expected number of tosses that will be required?

(b) What is the expected number of tails that will be obtained before the first head is obtained?

Short Answer

Expert verified

a) 2

b) 3

Step by step solution

01

Given information

A fair coin is tossed repeatedly until a head is obtained for the first time.

Let X be the variable

02

Calculate E(X)

\(\begin{array}{l}E\left( X \right)\\ = \sum\limits_x x \,\frac{1}{{{2^x}}}\\ = 2\end{array}\)

03

(b) Given the information

A fair coin is tossed repeatedly until a head is obtained for the first time.

Y is the expected number of tails obtained before the first head.

04

Calculate E(Y)

\(\begin{array}{c}E\left( Y \right) = \sum\limits_y y \,\frac{1}{{{3^y}}}\\ = 3\end{array}\)

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\({\bf{f}}\left( {\bf{x}} \right){\bf{ = }}\left\{ \begin{array}{l}\frac{{\bf{1}}}{{{\bf{2}}\left| {\bf{x}} \right|\left( {\left| {\bf{x}} \right|{\bf{ + 1}}} \right)}}{\bf{,x = \pm 1, \pm 2 \ldots ,}}\\{\bf{0,Otherwise}}\end{array} \right.\)

\({\bf{f}}\left( {\bf{x}} \right){\bf{ = }}\left\{ \begin{array}{l}\frac{{\bf{1}}}{{{\bf{x}}\left( {{\bf{x + 1}}} \right)}}{\bf{,x = 1,2,3 \ldots ,}}\\{\bf{0,Otherwise}}\end{array} \right.\)
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