91Ó°ÊÓ

Verify that each of the following functions is a probability density function. \(f(x)=\frac{1}{18} x, 0 \leq x \leq 6\)

Find the expected value and variance for each random variable whose probability density function is given. When computing the variance, use formula (5). \(f(x)=\frac{1}{18} x, 0 \leq x \leq 6\)

Let \(X\) be a Poisson random variable with parameter \(\lambda=5 .\) Compute the probabilities \(p_{0}, \ldots, p_{6}\) to four decimal places.

Verify that each of the following functions is a probability density function. \(f(x)=2(x-1), 1 \leq x \leq 2\)

Verify that each of the following functions is a probability density function. \(f(x)=\frac{1}{4}, 1 \leq x \leq 5\)

Find the expected value and variance for each random variable whose probability density function is given. When computing the variance, use formula (5). \(f(x)=\frac{1}{4}, 1 \leq x \leq 5\)

Verify that each of the following functions is a probability density function. \(f(x)=\frac{8}{9} x, 0 \leq x \leq \frac{3}{2}\)

Find the expected value and variance for each random variable whose probability density function is given. When computing the variance, use formula (5). \(f(x)=\frac{8}{9} x, 0 \leq x \leq \frac{3}{2}\)

Find the expected value and variance for each random variable whose probability density function is given. When computing the variance, use formula (5). \(f(x)=5 x^{4}, 0 \leq x \leq 1\)

Expected Value The number of accidents per week at a busy intersection was recorded for a year. There were 11 weeks with no accidents, 26 weeks with one accident, 13 weeks with two accidents, and 2 weeks with three accidents. A week is to be selected at random and the number of accidents noted. Let \(X\) be the outcome. Then, \(X\) is a random variable taking on the values \(0,1,2\), and 3 . (a) Write out a probability table for \(X\). (b) Compute \(\mathrm{E}(X)\). (c) Interpret \(\mathrm{E}(X)\).