A random variable X has a uniform density function $f(x)=\frac{1}{5}$ on $20 \leq x \leq 25$.

(a) Find E(X) and Var(X).

(b) Find b such that $\operatorname{Pr}(X \leq b)=.3$.

Give two ways to compute the variance of a continuous random variable.

A service contract on a computer costs $100 per year. The contract covers all necessary maintenance and repairs on the computer. Suppose that the actual cost to the manufacturer for providing this service is a random variable X (measured in hundreds of dollars) whose probability density function is$f(x)=(x-5)^i / 625,0 \leq x \leq 5$. Compute E(X) and determine how much money the manufacturer expects to make on each service contract on average.

Suppose that the weather forecast in previous Exercise indicates a 10% chance that cold weather will reduce the citrus grower's profit from $100,000 to$85,000 and a 10% chance that cold weather will reduce the profit to $75,000. Should the grower spend$5000 to protect the citrus fruit against the possible bad weather?

If X is a geometric random variable with parameter p=.9, compute the probabilities $p_0, \ldots, p_5$ and make a histogram.

Let X be the proportion of new restaurants in a given year that make a profit during their first year of operation, and suppose that the density function for X is $f(x)=20 x^3(1-x)$, $0 \leq x \leq 1$

(a) Find E(X) and give an interpretation of this quantity.

(b) Compute Var(X).

What is a probability table?

Find (by inspection) the expected values and variances of the exponential random variables with the density functions given in Exercise given below.

$$3 e^{-3 x}$$

Verify that each of the following functions is a probability density function.

$$f(x)=\frac{1}{18} x, 0 \leq x \leq 6$$

Let X be a continuous random variable on $0 \leq x \leq 2$, with the density function $f(x)=\frac{3}{8} x^2$.

(a) Calculate $\operatorname{Pr}(X \leq 1)$ and $\operatorname{Pr}(1 \leq X \leq 1.5)$.

(b) Find E(X) and Var(X).

Table 4 is the probability table for a random variable X. Find E(X), Var(X), and the standard deviation of X.

Suppose that a random variable X has a Poisson distribution with $\lambda=3$, as in Example mentioned. Compute the probabilities $p_6, p_7, p_8$.

Find the expected value and variance for each random variable whose probability density function is given. When computing the variance, use the said formula.

$$f(x)=\frac{1}{18} x, 0 \leq x \leq 6$$

Verify that each of the following functions is a probability density function.

$$f(x)=\frac{1}{18} x, 0 \leq x \leq 6$$