Exam 12: Probability and Calculus

A hardware store will cut lumber any length between 5 and 20 feet. Say X is the length of lumber requested by a customer. Then X is a uniform random variable with probability density function $f ( x ) = \frac { 1 } { 15 }$ Find E(X) and Var(X). Enter just two reduced fractions (unlabeled) in the order E(X), Var(X) separated by a comma.

The probability density function for a random variable X is f(x) = $3 x ^ { - 4 }$ , x ≥ 1. Find $\operatorname { Pr } ( 2 \leq X )$ . Enter just a reduced fraction.

The table below is the probability table for a random variable X. Find E(X). Outcome -1 0 1 2 Probability Enter just a reduced fraction of form $\frac { a } { b }$ .

It is estimated that the time between arrivals of visitors to a public library is an exponential random variable with expected value of 13 minutes. Find the probability that 30 minutes elapses without any arrivals. Enter your answer as just $\mathrm { e } ^ { \mathrm { a } / \mathrm { b } }$ , where $\frac { a } { b }$ is a reduced fraction.

A new car dealer observes that the number X of warranty claims for repairs on each new car sold is Poisson distributed, with an average of six claims per car. Compute the probability that a new car sold by the dealer will have no more than three warranty claims. Enter your answer in the form a $\mathrm { e } ^ { \mathrm { b } }$ .

Let X be a continuous random variable A ≤ X ≤ B and let f (x) be its probability density function and F (x) its cumulative distribution function. Indicate whether the following statements are true or false. - $F ^ { \prime }$ (x) = f(x)

A random variable X has probability density function f(x) = $k e ^ { - k x }$ (x ≥ 1) for some constant k. Suppose that $\operatorname { Pr } ( 1 \leq X \leq 2 ) = \frac { 1 } { 4 }$ , what is the value of k?

Find the expected value and variance for the random variable whose probability density function is $f ( x ) = 2 ( x - 1 )$ 1 ≤ x ≤ 2. Enter just two reduced fractions (unlabeled) in the order E(X), Var(X) separated by a comma.

The table below is the probability table for a random variable X. Find Var(X). Outcome -1 0 1 2 Probability Enter just a reduced fraction of form $\frac { a } { b }$ .

Find the expected value and variance for the random variable whose probability density function is $f ( x ) = 4 x - 1$ $\frac { 1 } { 2 }$ ≤ x ≤ 1. Enter just two reduced fractions (unlabeled) in the order E(X), Var(X) separated by a comma.

Find (by inspection) the expected value and the variance of the random variables with the following density function: f(x) = $\frac { 1 } { 4 \sqrt { 2 \pi } }$ $\mathrm { e } ^ { - ( 1 / 2 ) [ ( \mathrm { x } - 1 ) / 4 ] ^ { 2 } }$ , -∞ < x < ∞ Enter your answer as just two integers separated by a comma, the first representing E(X) and the second representing Var(X).

A random variable X has a cumulative distribution function F(x) = 1 - $\frac { 1 } { ( x + 1 ) ^ { 2 } }$ for $x \geq 0$ . Find $\operatorname { Pr } ( 1 \leq X \leq 4 )$ . Enter just a reduced fraction.