Suppose that if a signal value $s$ is sent from location A, then the signal value received at location B is normally distributed with parameters ($s$, 1). $S$ is normally distributed with parameters ($\mu$, $\sigma^2$). Let $S$ denote the signal sent and $R$ the signal received.
If $X_1, X_2, X_3, X_4, X_5$ are independent and identically distributed exponential random variables with the parameter $\lambda$, compute
Let $Y_1, Y_2, \cdot,Y_n$ denote a random sample from the uniform distribution $f(y) = 1, 0 \leq y \leq 1$. Find the probability density function for the range $R = Y_{(n)} − Y_{(1)}$
A.J. has 20 jobs that she must do in sequence, with the times required to do each of these jobs being independent random variables with mean 50 minutes and standard deviation 10 minutes. M.J. has 20 jobs that he must do in sequence, with the times required to do each of these jobs being independent random variables with mean 52 minutes and standard deviation 15 minutes.
If $Y1$ and $Y2$ are independent random variables, each having a normal distribution with mean 0 and variance 1, find the moment-generating function of $U = Y_1 Y_2$. Use this moment-generating function to find $\operatorname E(U)$ and $\operatorname V(U)$. Check the result by evaluating $\operatorname E(U)$ and $\operatorname V(U)$ directly from the density functions for $Y_1$ and $Y_2$.
The times to process orders at the service counter of a pharmacy are exponentially distributed with mean 10 minutes. If 100 customers visit the counter in a 2-day period, what is the probability that at least half of them need to wait more than 10 minutes?