The lifetime in hours of an electronic tube is a random variable having a probability density function given by $$ f(x)=\begin{cases} 0,& x<0,\\ xe^{-x},& x\geq 0, \end{cases} $$ Compute the expected lifetime of such a tube.
A point is chosen at random on a line segment of length $L$. Interpret this statement, and find the probability that the ratio of the shorter to the longer segment is less than $1/4$.
You arrive at a bus stop at 10 o'clock, knowing that the bus will
arrive at some time uniformly distributed between 10 and 10:30.
a. What is the probability that you will have to
wait longer than 10 minutes?
b. If, at 10:15, the bus has not yet arrived, what is the probability
that you will have to wait at least an additional 10 minutes?
The annual rainfall (in inches) in a certain region is normally distributed with $\mu=40$ and $\sigma=4$. What is the probability that, starting with this year, it will take over 10 years before a year occurs having a rainfall of over 50 inches? What assumptions are you making?
The number of years a radio functions is exponentially distributed with parameter $\lambda=\frac{1}{8}$. If Jones buys a used radio, what is the probability that it will be working after an additional 8 years?
Show that $$ E[Y]=\int _0^\infty P(Y>y)\, dy-\int_0^\infty P(Y<-y)\, dy. $$
Show that if $X$ has density function $f_X$, then $$ E\bigl[ g(X)\bigr]=\int _{-\infty}^\infty g(x)\, f_X(x)\, dx. $$
Use the result that, for a nonnegative random vari- able $Y$, $$ E[Y]=\int_0^\infty P(Y>t)\, dt $$ to show that, for a nonnegative random variable $X$, $$ E[X^n]=\int_0^\infty nx^{n-1}P(X>x)\, dx. $$
Show that, if $Z$ is a standard normal random variable,
then, for $x>0$
a. $P\bigl(Z>x\bigr)=P\bigl(Z<-x\bigr)$.
b. $P\bigl(|Z|>x\bigr)=2P\bigl(Z>x\bigr)$.
c. $P\bigl(|Z|<x\bigr)=2P\bigl(Z<x\bigr)-1$.
The median of a continuous random variable having distribution
function $F$ is that value $m$ such that $F(m)=1/2$ . That is, a random
variable is just as likely to be larger than its median as it is to be
smaller. Find the median of $X$ if $X$ is
a. uniformly distributed over an interval (a, b).
b. normal with parameters $\mu$ and $\sigma^2$.
c. exponential with rate $\lambda$.