Exam 12: Probability and Calculus

Suppose X is a normal random variable with density function $f ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { ( - 1 / 2 ) ( x + 4 ) ^ { 2 } }$ . Find the expected value and standard deviation of X.

Suppose that during a certain part of the day, the number X of automobiles that arrive within any one minute at a tollgate is Poisson distributed, and $\operatorname { Pr } ( X = \mathrm { k } ) = \frac { 4 ^ { \mathrm { k } } \mathrm { e } ^ { - 4 } } { 1 \cdot 2 \cdot \ldots \cdot \mathrm { k } }$ . What is the average number of automobiles that arrive per minute? Enter just an integer.

Find (by inspection) the expected value and the variance of the random variables with the following density function: $f ( x ) = 0.2 e ^ { - 0.2 x } , x \geq 0$ Enter your answer as just two integers separated by a comma, the first representing E(X) and the second representing Var(X).

Determine the probability of an outcome of the probability density function $f ( x ) = 4 x ^ { 3 }$ being between $\frac { 1 } { 4 } \text { and } \frac { 1 } { 2 }$ where $0 \leq x \leq 1$ . Enter just a reduced fraction.

Find the expected value and variance for the random variable whose probability density function is $f ( x ) = e ^ { x }$ $x \leq 0$ (You may use the fact that $\lim _ { b \rightarrow \infty }$ $b e^{b}$ = 0 .)Enter just two integers (unlabeled) in the order E(X), Var(X) separated by a comma.

An appliance comes with an unconditional money back guarantee for its first 6 months. It has been found that the time before the appliance experiences some sort of malfunction is an exponential random variable with mean 2 years. What percentage of appliances will malfunction during the warranty period? Enter your answer as just a ± $\mathrm { e } ^ { \mathrm { b } }$ , where a is an integer and b is a real number to two decimal places. (no units).

Find (by inspection) the expected value and standard deviation of the random variable with the following density function: $\mathrm { f } ( \mathrm { x } ) = \mathrm { e } ^ { - 0.1 x }$ . Enter your answer as just two integers separated by a comma, the first representing E(X) and the second representing $\sqrt { \operatorname { Var } ( X ) }$ .

Consider a square with sides of length 2 as in the diagram below. An experiment consists of choosing a point at random from the square and noting its x-coordinate. If X is the x-coordinate of the point chosen, find the cumulative distribution function of X. [Recall F(x) = Pr(0 ≤ X ≤ x).] Enter just an unlabeled polynomial in x in standard form.

The table below is the probability table for a random variable X. Find E(X), Var(X), and the standard deviation of X. Outcome -2 -1 0 1 2 Probability 0.2 0.35 0.15 0.05 0.25

Find (by inspection) the expected value and standard deviation of the random variable with the following density function: $f ( x ) = \frac { 1 } { 2 \sqrt { 2 \pi } } e ^ { - 1 / 8 x ^ { 2 } }$ Enter your answer as just two numbers (integers or reduced fractions) separated by a comma, the first representing E(X) and the second representing $\sqrt { \operatorname { Var } ( X ) }$ .

Is f(x) = $\frac { 1 } { 21 }$ $x ^ { 2 }$ a probability density function on the interval 1 ≤ x ≤ 4 ?

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(A) = 0, f(B) = 1

Find the expected value and variance for the random variable whose probability density function is $f ( x ) = \frac { 1 } { 3 } ,$ $2 \leq x \leq 5 .$ Enter just two reduced fractions of form $\frac { a } { b }$ (unlabeled) in the order E(X), Var(X) separated by a comma.

Find the value of k that makes f(x) = kx a probability function on the interval $1 \leq x \leq 2$ .

Is f(x) = $\frac { 1 } { ( x + 1 ) ^ { 2 } }$ is a probability density function for x ≥ 0? If so, find P(X ≥ 2). Enter either "no" or just a reduced fraction of form $\frac { a } { b }$ .

A random variable X has a cumulative distribution function F(x) = 1 - $\frac { 1 } { x ^ { 2 } }$ (x≥1). Find Pr(a ≤ X ≤ 5).

A carnival game costs $2 to play. A player draws a ball at random from a sack containing 1 white ball, 2 blue balls, 3 red balls, and 4 yellow balls. The payoff for drawing a particular color ball is as follows: white pays $5, blue pays $4, red pays $3 and yellow pays nothing. If X is the amount of money a player wins. Calculate E(X). Enter just a real number rounded off to two decimal places (no label).

A random variable X has a density function f(x) = $\frac { 1 } { \ln 16 }$ ∙ $\frac { 1 } { x }$ , 1 ≤ x ≤ 16. Find a such that $\operatorname { Pr } ( 1 \leq X \leq a ) = \frac { 3 } { 4 }$ . Enter just an integer, no labels.

Find the expected value of the random variable whose density function is $f ( x ) = \frac { 3 } { 8 } x ^ { 2 } , 0 \leq x \leq 2$ .

The table below is the probability table for a random variable X. Find E(X). Outcome -3 -2 -1 1 2 3 Probability 0.1 0.1 0.4 0.3 0.05 0.05 Enter just a real number rounded off to two decimal places.