Compute $E[X^2]$ for an exponential random variable $X$ with parameter $ \lambda$.
A bus travels between the two cities A and B, which are 100 miles apart. If the bus has a breakdown, the distance from the breakdown to city A has a uniform distribution over $(0,100)$. There is a bus service station in city A, in B, and in the center of the route between A and B. It is suggested that it would be more efficient to have the three stations located 25, 50, and 75 miles, respectively, from A. Do you agree? Why?
A man aiming at a target receives 10 points if his shot is within 1 inch of the target, 5 points if it is between 1 and 3 inches of the target, and 3 points if it is between 3 and 5 inches of the target. Find the expected number of points scored if the distance from the shot to the target is uniformly distributed between 0 and 10.
In 10,000 independent tosses of a coin, the coin landed on heads 5800 times. Is it reasonable to assume that the coin is not fair?
A model for the movement of a stock supposes that if the present price of the stock is $s$, then, after one period, it will be either $us$ with probability $p$ or $ds$ with probability $1-p$. Assuming that successive movements are independent, approximate the probability that the stock's price will be up at least 30% after the next 1000 periods if $u=1.012$, $d=0.990$, and $p=0.52$.
If $X$ is uniformly distributed over $(0,1)$, find the density function of $Y=e^X$.
Define a collection of events $E_a$ for $0<a<1$, having the property that $P(E_a)=1$ for all a but $$ P \bigl( \bigcap _a E_a\bigr) =0. $$
Let $X$ be a random variable that takes on values between $0$ and $c$. That is, $P(0\leq X\leq c)=1$. Show that $$ \operatorname{Var}(X)\leq \frac{c^2}{4} $$
If $X$ is an exponential random variable with mean $1/\lambda$, show that $$ E[X^k]=\frac{k!}{\lambda^k},\: k=1,2,\dots $$
If $X$ is an exponential random variable with parameter $ \lambda$, and $c>0$, show that $cX$ is exponential with parameter $ \lambda/c$.