Chapter 15: Q7P (page 749)

A weighted coin with probability p of coming down heads is tossed three times; x = number of heads minus number of tails.

Short Answer

Expert verified

The required values are mentioned below.

$\begin{matrix} μ (x) = 3 [2 p - 1] \\ var (x) = 12 p - 12 p^{2} \\ σ = 2 \sqrt{3 p (1 - p)} \end{matrix}$

Step by step solution

Step 1: Given Information

Three coins are tossed.

Definition of the cumulative distribution function.

The likelihood that a comparable continuous random variable has a value less than or equal to the function's argument is the value of the function.

Find the values.

Let S be the sample space.

$S = \{p^{3}, 3 p^{2} (1 - p), 3 p {(1 - p)}^{2}, {(1 - p)}^{3}\}$

Find the random variable.

$\begin{matrix} x_{1} = 3 \\ p_{1} = p^{3} \\ x_{2} = 1 \\ p_{2} = 3 p^{2} (1 - p) \end{matrix}$

Solve further.

$\begin{matrix} x_{3} = - 1 \\ p_{3} = 3 p {(1 - p)}^{2} \\ x_{4} = - 3 \\ p_{4} = {(1 - p)}^{3} \end{matrix}$

The mean is given below.

$\begin{matrix} μ (x) = \sum x_{i} p_{i} \\ = 3 p^{3} + 3 p^{2} (1 - p) - 3 p - 3 {(1 - p)}^{2} - 3 {(1 - p)}^{3} \\ = 3 [- p - 1 + 3 p] \\ = 3 [2 p - 1] \end{matrix}$

The variance is given below.

$\begin{matrix} var (x) = E (x^{2}) - μ^{2} \\ var (x) = 9 p^{3} + 3 p^{2} (1 - p) + 3 p {(1 - p)}^{2} + 9 {(1 - p)}^{3} - {[3 (2 p - 1)]}^{2} \\ var (x) = 9 p^{3} + 3 p^{2} - 3 p^{3} + 3 p (1 - 2 p + p^{2}) + 9 (1 - 3 p + 3 p^{2} - p^{3}) - 9 [4 p^{2} - 4 p + 1] \\ var (x) = 12 p - 12 p^{2} \end{matrix}$

The standard deviation is given below

$\begin{matrix} σ = \sqrt{var (x)} \\ σ = \sqrt{12 p - 12 p^{2}} \\ σ = 2 \sqrt{3 p (1 - p)} \end{matrix}$

The graph is shown below.

Hence, the required values are mentioned below.
$\begin{matrix} μ (x) = 3 [2 p - 1] \\ var (x) = 12 p - 12 p^{2} \\ σ = 2 \sqrt{3 p (1 - p)} \end{matrix}$

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A weighted coin with probability p of coming down heads is tossed three times; x = number of heads minus number of tails.

Short Answer

Step by step solution

Step 1: Given Information

Definition of the cumulative distribution function.

Find the values.

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