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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.5.35 (page 214)

The lung cancer hazard rate $λ (t)$ of a $t$ -year-old male smoker is such that
$λ (t) = . 027 + . 00025 {(t - 40)}^{2}$ $t \geq 40$
Assuming that a $40$ -year-old male smoker survives all other hazards, what is the probability that he survives to
$(a)$ age $50$ and $(b)$ age 60 without contracting lung cancer?

Short Answer

Expert verified

$t = 50 \Rightarrow λ (50) = 0.052$

$t = 60 \Rightarrow λ (60) = 0.127$

Step by step solution

01

Given Information.

$λ (t) = . 027 + . 00025 {(t - 40)}^{2} t \geq 40$

02

Part a Step 2 Explanation.

$t = 50,$

λ (50) = 0.027 + 0.00025 {(50 - 40)}^{2}

$= 0.027 + 0.025 = 0.052$

03

Part b Step 3 Explanation.

$t = 60, λ (60) = 0.0027 + 0.000259 {(60 - 40)}^{2}$

=0.127

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Most popular questions from this chapter

Verify that the gamma density function integrates to $1$ .

Show that $E [Y] = \int_{0}^{\infty} P \{Y > y\} d y - \int_{0}^{\infty} P \{Y < - y\} d y$ .

Hint: Show that $\int_{0}^{\infty} P \{Y < - y\} d y = - \int_{- \infty}^{0} x f_{y} (x) d x$

$\int_{0}^{\infty} P \{Y > y\} d y = \int_{0}^{\infty} x f_{y} (x) d x$

For some constant c, the random variable X has the probability density function:

$f (x) = \{\begin{cases} c x^{4} 0 < x < 2 \\ 0 otherwise \end{cases}$

Find

$E [X]$ and
$V a r (X)$

The time (in hours) required to repair a machine is an exponentially distributed random variable with parameters $λ = \frac{1}{2}$ . What is

$(a)$ the probability that a repair time exceeds $2$ hours?

$(b)$ the conditional probability that a repair takes at least $10$ hours, given that its duration exceeds $9$ hours?

Use the result that for a nonnegative random variable $Y$ ,

$E [Y] = \int_{0}^{\infty} P \{Y > t\} d t$

to show that for a nonnegative random variable $X$ ,

$E [X^{n}] = \int_{0}^{\infty} n x^{n - 1} P \{X > x\} d x$

Hint: Start with

$E [X^{n}] = \int_{0}^{\infty} P \{X^{n} > t\} d t$

and make the change of variables $t = x^{n}$ .

See all solutions

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