Chapter 15: Q1P (page 749)

Three coins are tossed; x = number of heads minus number of tails.

Short Answer

Expert verified

The required values are mentioned below.

$\begin{matrix} μ = 0 \\ var (x) = 3 \\ σ = \sqrt{3} \end{matrix}$ .

Step by step solution

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.

$S = {H H H, H H T, H T H, H T T, T H H, T H T, T H H, T T}$

Find the random variable.

Find the value of . $x_{1}$

$\begin{matrix} H = 0 \\ T = 3 \\ x_{1} = - 3 \\ p_{1} = 1 / 8 \end{matrix}$

Find the value of role="math" $x_{2}$ .

$\begin{matrix} H = 1 \\ T = 2 \\ x_{2} = - 1 \\ p_{2} = 3 / 8 \end{matrix}$

Find the value of $x_{3}$ .

$\begin{matrix} H = 2 \\ T = 1 \\ x_{3} = 1 \\ p_{1} = 3 / 8 \end{matrix}$

Find the value of $x_{4}$

$\begin{matrix} H = 3 \\ T = 0 \\ x_{4} = 3 \\ p_{4} = 1 / 8 \end{matrix}$

The mean is given below.

$\begin{matrix} μ = \sum x_{i} p_{i} \\ μ = - 3 \times \frac{1}{8} - \frac{3}{8} + \frac{3}{8} + 3 \times \frac{1}{8} \\ μ = 0 \end{matrix}$

The variance is given below.

$\begin{matrix} var (x) = \sum {(x_{i} - μ)}^{2} \\ p_{i} = \sum x_{i}^{2} p_{i} \\ = \times \frac{1}{8} \times \frac{3}{8} + \times \frac{3}{8} + \times \frac{1}{8} \\ var (x) = 3 \end{matrix}$

The standard deviation is given below.

$\begin{matrix} σ = \sqrt{var (x)} \\ σ = \sqrt{3} \end{matrix}$

Cumulative function is given below.

$\begin{matrix} x_{1} = - 3 \\ F (- 3) = 1 / 8 \\ x_{2} = - 1 \\ F (- 1) = 1 / 2 \end{matrix}$

Solve further.

$\begin{matrix} x_{3} = 1 \\ F (1) = 7 / 8 \\ x_{4} = 3 \\ F (3) = 1 \end{matrix}$

Thegraph is shown below.

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Three coins are tossed; x = number of heads minus number of tails.

Short Answer

Step by step solution

Given Information

Definition of the cumulative distribution function.

Find the values.

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