Question 1

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. 
-Pr(a ≤ X ≤ b) = $\int _ { a } ^ { b } f ( x ) d x$

Accepted Answer

The probability that a continuous random variable X falls within the interval [a, b] is given by the integral of its probability density function (pdf) f(x) from a to b, which is exactly what the provided equation represents.

Question 2

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 3

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.

Accepted Answer

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

Question 4

A random variable has probability density function f(x) = 30 $x ^ { 2 }$ (1 - x $)^2$ 
(0 ≤ x ≤ 1). Compute its cumulative distribution F(x).

Accepted Answer

The cumulative distribution function F(x) is the integral of the probability density function f(x) from 0 to x. Integrating f(x) = 30 $x ^ { 2 }$ (1 - x $)^2$ over this interval gives F(x) = 10 $x ^ { 3 }$ - 15 $x ^ { 4 }$ + 6 $x ^ { 5 }$.

Question 5

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 $x ^ { c }$ .

Accepted Answer

The answer of Suppose f(x) = \(\mathrm { kx }...

Question 6

Suppose f(x) = k( $x ^ { 2 }$ + 2x) is a probability density function for a continuous random variable on the interval $0 \leq x \leq 3$ Find the value of k and find the corresponding cumulative distribution function.
Enter just an unlabeled polynomial in x in standard form.

Accepted Answer

The answer of Suppose f(x) = k( \(x ^ {...

Question 7

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

The expected value $E(X)$ is calculated by integrating $x \cdot f(x)$ over the interval [0, 2]. The integral $\int_0^2 x \cdot \frac{3}{8} x^2 \, dx$ evaluates to $\frac{3}{2}$.

Question 8

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.

-Pr(A ≤ X ≤ b) = F(b)

Accepted Answer

Pr(A ≤ X ≤ b) = F(b) - F(a)

Question 9

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.

Accepted Answer

The answer of Missed work hours caused by one of...

Question 10

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)

Accepted Answer

The derivative of the cumulative distribution function $F(x)$ with respect to $x$ is equal to the probability density function $f(x)$, by definition.

Question 11

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.

-$\int _ { A } ^ { B } F ( x ) = 1$

Accepted Answer

The integral of the cumulative distribution function $F(x)$ from $A$ to $B$ does not equal 1. Instead, it is the integral of the probability density function $f(x)$ from $A$ to $B$ that equals 1, as this integral represents the total probability over the entire range of the continuous random variable.

Question 12

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 $\frac { a } { b }$ .

Accepted Answer

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

Question 13

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 14

A random variable X has a probability density function f(x) = $\frac { x } { 32 }$ , $0 \leq x \leq 8$ . Find a such that $\operatorname { Pr } ( X \geq a ) = \frac { 1 } { 4 }$ Enter your answer exactly in the reduced form b $\sqrt { c }$ , unlabeled.

Accepted Answer

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

Question 15

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

We cannot say that f(A) = 0 and f(B) = 1 in general. It is possible for the probability density function to have positive values at A and/or B, and it is also possible for the area under the curve to be less than 1. However, we do know that F(A) = 0 and F(B) = 1, since the cumulative distribution function always starts at 0 and ends at 1 for any continuous random variable.

Question 16

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).

Accepted Answer

The answer of Missed work hours caused by one of...

Question 17

Given the probability density function f(x) = $\frac { 1 } { 3 }$ , determine the corresponding cumulative distribution function where 12 ≤ x ≤ 15.
Enter an unlabeled polynomial in x in standard form.

Accepted Answer

The answer of Given the probability density function f(x) =...

Question 18

Given the density function f(x) = $\frac { 3 } { 64 }$ $x ^ { 2 }$ , 0 ≤ x ≤ 4, determine the corresponding cumulative distribution function.
Enter just an unlabeled polynomial in x in standard form.

Accepted Answer

The answer of Given the density function f(x) = \(\frac...

Question 19

Find the expected value and variance for the random variable whose probability density function is $f ( x ) = x$ $0 \leq x \leq 1 .$ Enter just two reduced fractions (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 20

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.

Accepted Answer

The answer of Missed work hours caused by one of...

Quiz 12: Probability and Calculus

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. -Pr(a ≤ X ≤ b) = $\int _ { a } ^ { b } f ( x ) d x$

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 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.

A random variable has probability density function f(x) = 30 $x ^ { 2 }$ (1 - x $) ^2$ (0 ≤ x ≤ 1) . Compute its cumulative distribution F(x) .

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 $x ^ { c }$ .

Suppose f(x) = k( $x ^ { 2 }$ + 2x) is a probability density function for a continuous random variable on the interval $0 \leq x \leq 3$ Find the value of k and find the corresponding cumulative distribution function. Enter just an unlabeled polynomial in x in standard form.

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

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. -Pr(A ≤ X ≤ b) = F(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)

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. - $\int _ { A } ^ { B } F ( x ) = 1$

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 $\frac { a } { b }$ .

A random variable X has a probability density function f(x) = $\frac { x } { 32 }$ , $0 \leq x \leq 8$ . Find a such that $\operatorname { Pr } ( X \geq a ) = \frac { 1 } { 4 }$ Enter your answer exactly in the reduced form b $\sqrt { c }$ , unlabeled.

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

Given the probability density function f(x) = $\frac { 1 } { 3 }$ , determine the corresponding cumulative distribution function where 12 ≤ x ≤ 15. Enter an unlabeled polynomial in x in standard form.

Given the density function f(x) = $\frac { 3 } { 64 }$ $x ^ { 2 }$ , 0 ≤ x ≤ 4, determine the corresponding cumulative distribution function. Enter just an unlabeled polynomial in x in standard form.

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

Quiz 12: Probability and Calculus

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. -Pr(a ≤ X ≤ b) = ∫abf(x)dx\int _ { a } ^ { b } f ( x ) d x∫ab​f(x)dx

Find the expected value and variance for the random variable whose probability density function is f(x)=3x2f ( x ) = 3 x ^ { 2 }f(x)=3x2 0 ≤ x ≤ 1. Enter just two reduced fractions (unlabeled) in the order E(X), Var(X) separated by a comma.

A random variable has probability density function f(x) = 30 x2x ^ { 2 }x2 (1 - x ) 2) ^2) 2 (0 ≤ x ≤ 1) . Compute its cumulative distribution F(x) .

Suppose f(x) = kx−5\mathrm { kx } ^ { - 5 }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 xcx ^ { c }xc .

Suppose f(x) = k( x2x ^ { 2 }x2 + 2x) is a probability density function for a continuous random variable on the interval 0≤x≤30 \leq x \leq 30≤x≤3 Find the value of k and find the corresponding cumulative distribution function. Enter just an unlabeled polynomial in x in standard form.

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 .

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. -Pr(A ≤ X ≤ b) = F(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′F ^ { \prime }F′ (x) = f(x)

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. - ∫ABF(x)=1\int _ { A } ^ { B } F ( x ) = 1∫AB​F(x)=1

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

Given the probability density function f(x) = 13\frac { 1 } { 3 }31​ , determine the corresponding cumulative distribution function where 12 ≤ x ≤ 15. Enter an unlabeled polynomial in x in standard form.

Given the density function f(x) = 364\frac { 3 } { 64 }643​ x2x ^ { 2 }x2 , 0 ≤ x ≤ 4, determine the corresponding cumulative distribution function. Enter just an unlabeled polynomial in x in standard form.

Find the expected value and variance for the random variable whose probability density function is f(x)=xf ( x ) = xf(x)=x 0≤x≤1.0 \leq x \leq 1 .0≤x≤1. Enter just two reduced fractions (unlabeled) in the order E(X), Var(X) separated by a comma.

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. -Pr(a ≤ X ≤ b) = $\int _ { a } ^ { b } f ( x ) d x$

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.

A random variable has probability density function f(x) = 30 $x ^ { 2 }$ (1 - x $) ^2$ (0 ≤ x ≤ 1) . Compute its cumulative distribution F(x) .

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 $x ^ { c }$ .

Suppose f(x) = k( $x ^ { 2 }$ + 2x) is a probability density function for a continuous random variable on the interval $0 \leq x \leq 3$ Find the value of k and find the corresponding cumulative distribution function. Enter just an unlabeled polynomial in x in standard form.

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

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)

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. - $\int _ { A } ^ { B } F ( x ) = 1$

Given the probability density function f(x) = $\frac { 1 } { 3 }$ , determine the corresponding cumulative distribution function where 12 ≤ x ≤ 15. Enter an unlabeled polynomial in x in standard form.

Given the density function f(x) = $\frac { 3 } { 64 }$ $x ^ { 2 }$ , 0 ≤ x ≤ 4, determine the corresponding cumulative distribution function. Enter just an unlabeled polynomial in x in standard form.

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