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A First Course in Probability

Sheldon M. Ross

$Math Studyset Vaia Explanations$ Math

9th Edition · 432 pages

ISBN: 9780321794772

Chapter 5: Q.5.18 (page 213)

Suppose that X is a normal random variable with mean $5$ . If $P (X > 9) = . 2$ , approximately what is $V a r (X)$ ?

Short Answer

Expert verified

The required variance is 22.66.

Step by step solution

01

Step 1. Given Information.

$X$ is a random variable with mean $5$ .

Also, $P (X > 9) = 0.2 .$

02

Step 2. Find VarX.

Here,

$P (X > 9) = 0.2 P (\frac{X - μ}{σ} > \frac{9 - μ}{σ}) = 0.2 1 - φ (\frac{9 - μ}{σ}) = 0.2 φ (\frac{9 - μ}{σ}) = 0.8 \frac{9 - μ}{σ} = 0.84 σ = \frac{9 - μ}{0.84} σ = \frac{9 - 5}{0.84} σ = 4.76 σ^{2} = {(4.76)}^{2} σ^{2} = 22.66$

Hence, Var $(X)$ =22.66.

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Most popular questions from this chapter

Use the identity of Theoretical Exercise 5.5 to derive E[X2] when X is an exponential random variable with parameter λ.

If $65$ percent of the population of a large community is in favor of a proposed rise in school taxes, approximate the probability that a random sample of $100$ people will contain

(a) at least $50$ who are in favor of the proposition;

(b) between $60$ and $70$ inclusive who are in favor;

(c) fewer than $75$ in favor.

Let be a random variable with probability density function

$f (x) = \{\begin{cases} c (1 - x^{2}) & - 1 < x < 1 \\ 0 & o t h e r w i s e \end{cases}$

(a) What is the value of?

(b) What is the cumulative distribution function of?

5.6. Compute $E [X]$ if $X$ has a density function given by

$(a) f (x) = \{\begin{cases} \frac{1}{4} x e^{\frac{- x}{2}} & x > 0 \\ 0 & o t h e r w i s e \end{cases}$ ;

$(b) f (x) = \{\begin{cases} c (1 - x^{2}) & - 1 < x < 1 \\ 0 & o t h e r w i s e \end{cases}$ ;

$(c) f (x) = \{\begin{cases} \frac{5}{x^{2}} & x > 5 \\ 0 & x \leq 5 \end{cases}$ .

Let $X$ be a random variable with probability density function

$f (x) = \{\begin{cases} c (1 - x^{2}) - 1 < x < 1 \\ 0 otherwise \end{cases}$

(a) What is the value of $c$ ?

(b) What is the cumulative distribution function of $X$ ?

See all solutions

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