Exam 12: Probability and Calculus

Find the expected value and variance for the random variable whose probability density function is $f ( x ) = 12 x ^ { 2 } ( 1 - x )$ 0 ≤ x ≤ 1. Enter just two reduced fractions (unlabeled) in the order E(X), Var(X) separated by a comma.

Suppose X is a random variable whose probabilities are Poisson distributed with $\mathrm { P } _ { \mathrm { n } }$ = $\frac { ( 14 ) ^ { n } } { n ! }$ $e ^ { - 14 }$ . Which of the following is true?

In a certain office, the number of typewriters that break down during any given week is Poisson distributed with λ = 2. What is the probability that more than three typewriters break down during a week? Enter your answer in the form $a \pm \frac { b } { c } e ^ { d }$ where $\frac { b } { c }$ is reduced.

Find the expected value and variance for the random variable whose probability density function is $f ( x ) = 3 x ^ { 2 }$ 0 ≤ x ≤ 1. Enter just two reduced fractions (unlabeled) in the order E(X), Var(X) separated by a comma.

The life of a battery is a random variable with probability density function $f ( x ) = \frac { 3 } { 56 } x ^ { 2 }$ , $2 \leq x \leq 4$ where x is the time in months. Calculate E(X). Enter just a reduced fraction of form $\frac { a } { b }$ unlabeled.

A random variable X is exponentially distributed with a mean of 10. Determine a so that $\operatorname { Pr } ( 0 \leq x \leq a ) = 0.75$

Let X be a continuous random variable with a cumulative distribution function $\mathrm { F } ( \mathrm { x } ) = 1 - \mathrm { e } ^ { - \mathrm { x } ^ { 2 } } ( \mathrm { x } \geq 0 )$ . Find $\operatorname { Pr } ( 1 \leq X \leq 2 )$

When a road crew inspects a road that hasn't been worked on for several years, then the distance between necessary repairs is an exponential random variable with a mean of 0.25 miles. What is the probability that the crew will find a mile long stretch of road that does not need repairs? Enter your answer as just $\mathrm { e } ^ { \mathrm {b} }$ .

A random variable X has a cumulative distribution function F(x) = $\frac { x } { 5 }$ - 2, $10 \leq x \leq 15$ . Find a such that $\operatorname { Pr } ( a \leq X \leq 15 ) = \frac { 2 } { 3 }$ Enter just a reduced fraction of form $\frac { a } { b }$ .

Missed work hours caused by one of a class of industrial accidents has a probability density function $f ( t ) = \frac { 1 } { 8 } e ^ { - t } + \frac { 3 } { 8 } e ^ { - t / 2 } + \frac { 1 } { 24 } e ^ { - t / 3 }$ where t is measured in hours. -What proportion of these accidents result in 5 or fewer missed work hours? Enter just a real number to two decimal places.

Suppose a small amount of blood is sampled and the number of white blood cells are counted. If the number of white blood cells is Poisson distributed with λ = 6, what is the probability that the sample has more than 4 white blood cells? What is the average number of white blood cells per sample? Is $\operatorname { Pr } ( 4 \leq X ) = 0.7149 ; E ( X ) = 6$ correct?

The table below is the probability table for a random variable X. Find E(X), Var(X), and the standard deviation of X. Outcome 40 50 60 70 80 Probability 0.3 0.15 0.15 0.2 0.2 Enter just three real numbers all rounded off to two decimal places: a, b, c representing the three quantities in the order requested above, separated by commas (no labels).

Missed work hours caused by one of a class of industrial accidents has a probability density function $f ( t ) = \frac { 1 } { 8 } e ^ { - t } + \frac { 3 } { 8 } e ^ { - t / 2 } + \frac { 1 } { 24 } e ^ { - t / 3 }$ where t is measured in hours. -What proportion of these accidents result in more than 9 missed work hours? Enter just a real number to two decimal places.

Missed work hours caused by one of a class of industrial accidents has a probability density function $f ( t ) = \frac { 1 } { 8 } e ^ { - t } + \frac { 3 } { 8 } e ^ { - t / 2 } + \frac { 1 } { 24 } e ^ { - t / 3 }$ where t is measured in hours. -Dr. Smith's test score distribution is characterized by the probability density function $f ( x ) = \frac { x \left( 10,000 - x ^ { 2 } \right) } { 25,000,000 }$ , $0 \leq x \leq 100$ . What percentage of people are likely to get a 60 or above on the exam? Enter just a real number to two decimal places (no units).

Is f(x) = $\left( \frac { 3 } { 2 } \right)$ x - 1 a probability density function for 0 ≤ x ≤ 2?

Find the value of k that makes f(x) = k $\sqrt { x }$ a probability density function on the interval $4 \leq x \leq 9$ Enter just a reduced fraction.

Suppose that a bag holds 3 blue balls and one red ball. We pull a ball from the bag at random, return it and then repeat the process. Suppose we continue pulling balls until the blue ball is drawn and then we observe the number of consecutive red balls drawn. What is the average number of red balls between occurrences of blue balls? Is $E ( X ) = \sum _ { n = 1 } ^ { \infty } n \frac { 3 { n } } { 4 ^ { n + 1 } } = \frac { 1 } { 4 } \sum _ { n = 1 } ^ { \infty } n \left( \frac { 3 } { 4 } \right) ^ { n } = 3$ correct?

A student taking five courses keeps a record of the number of assignments due each day in all her courses. Over the course of the 60-day semester she finds on 20 days no assignments are due, on 15 days an assignment is due in one course, on 15 days an assignment is due in two courses, on 9 days an assignments is due in three courses and once during the semester she has an assignment due in 4 courses. If X is the number of assignments due on a day selected at random from the semester, find E(X). Is $E ( X ) = 0 \cdot \frac { 1 } { 3 } + 1 \cdot \frac { 1 } { 4 } + 2 \cdot \frac { 1 } { 4 } + 3 \cdot \frac { 3 } { 20 } + 4 \cdot \frac { 1 } { 60 }$ the correct answer?

If f(x) = 6x(1 - x) is a probability density function for 0 ≤ x ≤ 1, find F(x), the corresponding cumulative distribution function and use it to find Pr $\left( \frac { 1 } { 2 } \leq X \leq 1 \right)$ . Enter just a reduced fraction representing $\operatorname { Pr } \left( \frac { 1 } { 2 } \leq X \leq 1 \right)$ Do not label.

Suppose f(x) = $\mathrm { kx } ^ { - 5 }$ is a density function for a random variable x for x ≥ 2. Find the value of k and find the corresponding cumulative distribution function. Enter your answer exactly as a ± b $x ^ { c }$ .