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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. 5.19 (page 213)

Let X be a normal random variable with mean $12$ and variance $4$ . Find the value of $c$ such that localid="1646649699736" $P \{X > c\} = 0.10$ .

Short Answer

Expert verified

The value of $c$ is $14.56$

Step by step solution

01

Step 1. Given information.

$X$ is a normal random variable with meanlocalid="1646649744005" $(μ) = 12$ and variance ${(σ)}^{2} = 4$

02

Step 2. Find value of c.

As given in the question,

Mean $(μ) = 12$ and Variance $σ^{2} = 4$ .

Hence, Standard Deviation $(σ) = 2$ .

As required,

$P [X > c] = 0.10 1 - P [X \leq c] = 1 - 0.10 P [X \leq c] = 0.90 P [\frac{X - M}{σ} \leq \frac{c - M}{σ}] = 0.90 P [Z \leq \frac{c - 12}{2}] = 0.90 \frac{c - 12}{2} = 1.28 c = (1.28 \times 2) + 12 c = 14.56$

Hence, the required value of $c$ is $14.56$ .

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Most popular questions from this chapter

A system consisting of one original unit plus a spare

can function for a random amount of time $X$ . If the density

of $X$ is given (in units of months) by

$f (x) = \{\begin{cases} C x e^{- x / 2} & x > 0 \\ 0 & x \leq 0 \end{cases}$

what is the probability that the system functions for at least $5$ months?

For some constant c, the random variable X has the probability density function f(x) = c x n 0 < x < 1 0 otherwise Find (a) c and

(b) P{X > x}, 0 < x < 1.

Two types of coins are produced at a factory: a fair coin and a biased one that comes up heads $55$ percent of the time. We have one of these coins but do not know whether it is a fair coin or a biased one. In order to ascertain which type of coin we have, we shall perform the following statistical test: We shall toss the coin $1000$ times. If the coin lands on heads $525$ or more times, then we shall conclude that it is a biased coin, whereas if it lands on heads fewer than $525$ times, then we shall conclude that it is a fair coin.

The salaries of physicians in a certain specialty are approximately normally distributed. If $25$ percent of these

physicians earn less than $\(180, 000$ and $25$ percent earn more than $\) 320, 000$ , approximately what fraction earn

(a) less than $\(200, 000$ ?

(b) between $\) 280, 000 and $ 320, 000$ ?

If X is an exponential random variable with parameter λ, and c > 0, show that cX is exponential with parameter λ/c

See all solutions

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