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.
Choose a random integer $X$ from the interval [0, 4]. Then choose a random integer $Y$ from the interval [0, $x$], where $x$ is the observed value of $X$. Make assumptions about the marginal pmf $f_X(x)$ and the conditional pmf $h(y | x)$ and compute $P(X + Y > 4)$.
Suppose that we have the following information about basketball player shoe size $S$, and player height $H$.
| H = Height | ||||
| 68 | 70 | 73 | ||
|---|---|---|---|---|
| S = Shoe Size | 8.5 | 0.25 | 0.20 | 0.15 |
| 12 | 0.05 | 0.12 | ? | |
| Find the expected shoe size, given that a player is 73 inches tall, $\operatorname E \lbracket S | H = 73 \rbracket$. |
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 $Y_1, Y_2, \ldots,Y_n$ denote a random sample from the uniform distribution $f(y) = 1, 0 \leq y \leq 1$. Find the probability density function for the range $R = Y_{(n)} − Y_{(1)}$
Let $X_1, X_2, \ldots, X_n$ be a random sample from a $\operatorname U(0, 1)$ population. Find the distribution of the sample range $X_{(n)} - X_{(1)}$.
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 $Y_1$ and $Y_2$ 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 $Y_2$.
A charity receives 2025 contributions. Contributions are assumed to be independent and identically distributed with mean 3125 and standard deviation 250. Calculate the approximate 90th percentile for the distribution of the total contributions received.
What is the expected number of rolls of a fair die required to see all six outcomes? What is the population variance of the number of rolls of a fair die required to see all six outcomes?
The number of trees in one acre has a Poisson distribution with mean 60. Assuming independence, compute $P(5950 \leq X \leq 6100)$, approximately, where $X$ is the number of trees in 100 acres.
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?
A driver and a passenger are in a car accident. Each of them independently has probability 0.3 of being hospitalized. When a hospitalization occurs, the loss is uniformly distributed on [0, 1]. When two hospitalizations occur, the losses are independent. Calculate the expected number of people in the car who are hospitalized, given that the total loss due to hospitalizations from the accident is less than 1.
Cassie is interested in determining the average gas mileage for a fleet of a particular model of car. Previous data collection of gas mileage gives her confidence that the standard deviation of the gas mileages that she collects will have a standard deviation of 3.3 miles per gallon. How many cars should she sample in order to be at least 95% certain that the difference between the population mean and sample mean is less than 1 mile per gallon?