91Ó°ÊÓ

If $X$is an exponential random variable with a parameter$\mathrm{λ}=1$, compute the probability density function of the random variable $Y$defined by $Y=\log \left(X\right)$

The probability density function of X, the lifetime of a certain type of electronic device (measured in hours), is given by

$f\left(x\right)=\left\{\begin{array}{ll}\frac{10}{x^2}& x>10\\ 0& x≤10\end{array}\right.$

$\left(a\right)$Find$P\{X>20\}$

localid="1646589462481" $\left(b\right)$What is the cumulative distribution function of localid="1646589521172" $X?$

localid="1646589534997" $\left(c\right)$) What is the probability that of$6$such types of devices, at least localid="1646589580632" $3$will function for at least localid="1646589593287" $15$hours? What assumptions are you making?

The random variable $X$has the probability density function

$f\left(x\right)=\left\{\begin{array}{ll}ax+bx2& 0<x<1\\ 0& otherwise\end{array}\right.$

If $E\left[X\right]=6$, find

(a) $P\{X<\frac{1}{2}\}$and

(b) $Var\left(X\right)$.

Prove Corollary$2.1$.

If $X$is uniformly distributed over$(0,1)$, find the density function of$Y=e^X$.

Find the distribution of$R=A\sin \left(\mathrm{θ}\right)$, where $A$is a fixed constant and $\mathrm{θ}$is uniformly distributed on$\left(\raisebox{1ex}{$-\mathrm{Ï€}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,\raisebox{1ex}{$\mathrm{Ï€}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)$. Such a random variable $R$arises in the theory of ballistics. If a projectile is fired from the origin at an angle $\mathrm{Î\pm }$from the earth with a speed$\mathrm{ν}$, then the point $R$at which it returns to the earth can be expressed as$R=\left(\raisebox{1ex}{$v^2$}\!\left/ \!\raisebox{-1ex}{$g$}\right.\right)\sin \left(2\mathrm{Î\pm }\right)$, where $g$is the gravitational constant, equal to $980$centimeters per second squared.

Let $Y$be a lognormal random variable (see Example 7e for its definition) and let $c>0$be a constant. Answer true or false to the following, and then give an explanation for your answer.

(a) $cY$is lognormal;

(b) $c+Y$is lognormal.

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

$E\left[Y\right]={∫}_0^{∞}P\left\{Y>t\right\}dt$

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

$E\left[X^n\right]={∫}_0^{∞}n{x}^{n-1}P\left\{X>x\right\}dx$

Hint: Start with

$E\left[X^n\right]={∫}_0^{∞}P\left\{X^n>t\right\}dt$

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

The random variable X is said to be a discrete uniform random variable on the integers 1, 2, . . . , n if P{X = i } = 1 n i = 1, 2, . . , n For any nonnegative real number x, let In t(x) (sometimes written as [x]) be the largest integer that is less than or equal to x. Show that if U is a uniform random variable on (0, 1), then X = In t (n U) + 1 is a discrete uniform random variable on 1, . . . , n.

A filling station is supplied with gasoline once a week. If its weekly volume of sales in thousands of gallons is a random variable with probability density function

$f\left(x\right)=\left\{\begin{array}{ll}5{(1−x)}^4& 0<x<1\\ 0& otherwise\end{array}\right.$

what must the capacity of the tank be so that the probability of the supply being exhausted in a given week is role="math" localid="1646634562935" $.01?$