Chapter 15: Q13P (page 765)

Generalize problem $12$ to show that for the general binomial distribution $(6.15)$ , $σ_{x =} \sqrt{n p q}$ and $V a r (x) = n p q$

Short Answer

Expert verified

Result variance and standard deviation are mentioned below:

V a r (x) = n p q σ = \sqrt{n p q}

Step by step solution

Given Information

$p (x = 0) = p p (x = 1) = q p + q = 1$

Definition of Variance and Standard deviations

The standard deviation of a bunch of numbers is the distance between them and the mean. The variance is a measure of how much each point deviates from the mean on average.

Calculate expected value

Find expected value.

$\begin{array}{rcl} μ_{x} & = & \underset{}{\sum x_{i} p (x_{i})} \\ = & 1 \times p + 0 \times q \\ = & p \end{array}$

Calculate variance

Find variance.

$\begin{array}{rcl} V a r (x) & = & \sum_{1} {(x - μ)}^{2} p (x_{i}) \\ = & (1 - p^{2}) p + {(0 - p)}^{2} q \\ = & q^{2} p + p^{2} q \\ = & q p (p + q) \end{array}$

$\begin{array}{rcl} V a r (x) & = & p q \end{array}$

Calculate variance and standard deviation for nth experiment

Solve for variance.

$\begin{array}{rcl} V a r (x) & = & V a r (x_{1} + x_{2} + x_{3} + . . . + x_{n}) \\ = & V a r (x_{1}) + V a r (x_{2}) + V a r (x_{3}) + . . . + V a r (x_{n}) \\ = & V a r (x) + V a r (x) + V a r (x) + . . . + V a r (x) \\ = & n V a r (x) \\ V a r (x) & = & n p q \end{array}$

Solve for standard deviation.

$\begin{array}{rcl} σ & = & \sqrt{n p q} \end{array}$

Hence, variance and standard deviation are mentioned below:

$\begin{array}{rcl} V a r (x) & = & n p q \\ σ & = & \sqrt{n p q} \end{array}$

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Generalize problem $12$ to show that for the general binomial distribution $(6.15)$ , $σ_{x =} \sqrt{n p q}$ and $V a r (x) = n p q$

Short Answer

Step by step solution

Given Information

Definition of Variance and Standard deviations

Calculate expected value

Calculate variance

Calculate variance and standard deviation for nth experiment

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Most popular questions from this chapter

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Generalize problem 12to show that for the general binomial distribution (6.15) ,σx=npqandVar(x)=npq

Short Answer

Step by step solution

Given Information

Definition of Variance and Standard deviations

Calculate expected value

Calculate variance

Calculate variance and standard deviation for nth experiment

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fields in Physics

Dynamics

Measurements

Oscillations

Atoms and Radioactivity

Translational Dynamics

Study anywhere. Anytime. Across all devices.

Generalize problem $12$ to show that for the general binomial distribution $(6.15)$ , $σ_{x =} \sqrt{n p q}$ and $V a r (x) = n p q$