Question 1

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.

Accepted Answer

A)  μ = -4, σ = 1 
B)  μ = 1, σ = 4 
C)  μ = 1, σ = 16 
D)  μ = 4, σ = 1 
E)  μ = -4, σ = 16

Question 2

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.

Accepted Answer

The answer of Suppose that during a certain part of...

Question 3

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).

Accepted Answer

The answer of Find (by inspection) the expected value and...

Question 4

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.

Accepted Answer

The answer of Determine the probability of an outcome of...

Question 5

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.

Accepted Answer

The answer of Find the expected value and variance for...

Question 6

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).

Accepted Answer

The answer of An appliance comes with an unconditional money...

Question 7

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 $\sqrt { \operatorname { Var } ( X ) }$ .

Accepted Answer

The answer of Find (by inspection) the expected value and...

Question 8

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.

Accepted Answer

The answer of Consider a square with sides of length...

Question 9

The table below is the probability table for a random variable X. Find E(X), Var(X), and the standard deviation of X. $$\begin{array} { l | l l l l l } 
\text { Outcome } & - 2 & - 1 & 0 & 1 & 2 \
\hline \text { Probability } & 0.2 & 0.35 & 0.15 & 0.05 & 0.25
\end{array}$$

Accepted Answer

A)  E(X) = -0.2; Var(X) = 2.22; standard deviation of X =1.49 
B)  E(X) = -0.2; Var(X) = 2.16; standard deviation of X =1.47 
C)  E(X) = -0.2; Var(X) = 4; standard deviation of X =2 
D)  E(X) = -0.2; Var(X) = 0; standard deviation of X = 0 
E)  none of these

Question 10

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 $\sqrt { \operatorname { Var } ( X ) }$ .

Accepted Answer

The answer of Find (by inspection) the expected value and...

Question 11

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

Accepted Answer

A) True 
 B)False

Question 12

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

Accepted Answer

A) True 
 B)False

Question 13

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.

Accepted Answer

The answer of Find the expected value and variance for...

Question 14

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

Accepted Answer

A)  $\frac { 4 } { 5 }$ 
B)  $\frac { 2 } { 5 }$ 
C)  $\frac { 2 } { 3 }$ 
D)  $\frac { 6 } { 5 }$ 
E)  none of these

Question 15

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 $\frac { a } { b }$ .

Accepted Answer

The answer of Is f(x) = \(\frac { 1 }...

Question 16

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

Accepted Answer

A)  1 - $\frac { 1 } { a ^ { 2 } }$ 
B)  $\frac { 24 } { 25 }$ - $\frac { 1 } { a ^ { 2 } }$ 
C)  $\frac { 1 } { a ^ { 2 } }$ - $\frac { 1 } { 25 }$ 
D)  $\frac { 1 } { a ^ { 2 } }$ 
E)  none of these

Question 17

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).

Accepted Answer

The answer of A carnival game costs $2 to play....

Question 18

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.

Accepted Answer

The answer of A random variable X has a density...

Question 19

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

Accepted Answer

A)  $\frac { 5 } { 2 }$ 
B)  $\frac { 5 } { 8 }$ 
C)  $\frac { 3 } { 2 }$ 
D)  $\frac { 3 } { 8 }$ 
E)  none of these

Question 20

The table below is the probability table for a random variable X. Find E(X). $$\begin{array} { l | l l l l l l } 
\text { Outcome } & - 3 & - 2 & - 1 & 1 & 2 & 3 \
\hline \text { Probability } & 0.1 & 0.1 & 0.4 & 0.3 & 0.05 & 0.05
\end{array}$$ Enter just a real number rounded off to two decimal places.

Accepted Answer

The answer of The table below is the probability table...

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.

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.

The table below is the probability table for a random variable X. Find E(X) , Var(X) , and the standard deviation of X. $\begin{array} { l | l l l l l } \text { Outcome } & - 2 & - 1 & 0 & 1 & 2 \\\hline \text { Probability } & 0.2 & 0.35 & 0.15 & 0.05 & 0.25\end{array}$

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 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). $\begin{array} { l | l l l l l l } \text { Outcome } & - 3 & - 2 & - 1 & 1 & 2 & 3 \\\hline \text { Probability } & 0.1 & 0.1 & 0.4 & 0.3 & 0.05 & 0.05\end{array}$ Enter just a real number rounded off to two decimal places.

Exam 12: Probability and Calculus

Suppose X is a normal random variable with density function f(x) =12πe(−1/2) (x+4) 2f ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { ( - 1 / 2 ) ( x + 4 ) ^ { 2 } }f(x) =2π​1​e(−1/2) (x+4) 2 . Find the expected value and standard deviation of X.

Determine the probability of an outcome of the probability density function f(x)=4x3f ( x ) = 4 x ^ { 3 }f(x)=4x3 being between 14 and 12\frac { 1 } { 4 } \text { and } \frac { 1 } { 2 }41​ and 21​ where 0≤x≤10 \leq x \leq 10≤x≤1 . Enter just a reduced fraction.

Is f(x) = 121\frac { 1 } { 21 }211​ x2x ^ { 2 }x2 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 value of k that makes f(x) = kx a probability function on the interval 1≤x≤21 \leq x \leq 21≤x≤2 .

Is f(x) = 1(x+1)2\frac { 1 } { ( x + 1 ) ^ { 2 } }(x+1)21​ is a probability density function for x ≥ 0? If so, find P(X ≥ 2). Enter either "no" or just a reduced fraction of form ab\frac { a } { b }ba​ .

A random variable X has a cumulative distribution function F(x) = 1 - 1x2\frac { 1 } { x ^ { 2 } }x21​ (x≥1) . Find Pr(a ≤ X ≤ 5) .

A random variable X has a density function f(x) = 1ln⁡16\frac { 1 } { \ln 16 }ln161​ ∙ 1x\frac { 1 } { x }x1​ , 1 ≤ x ≤ 16. Find a such that Pr⁡(1≤X≤a)=34\operatorname { Pr } ( 1 \leq X \leq a ) = \frac { 3 } { 4 }Pr(1≤X≤a)=43​ . Enter just an integer, no labels.

Find the expected value of the random variable whose density function is f(x) =38x2,0≤x≤2f ( x ) = \frac { 3 } { 8 } x ^ { 2 } , 0 \leq x \leq 2f(x) =83​x2,0≤x≤2 .

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.

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.

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

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 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 $f ( x ) = \frac { 3 } { 8 } x ^ { 2 } , 0 \leq x \leq 2$ .