Question 1

What is a random variable?

Accepted Answer

A random variable is a numerical variable whose value depends on the outcome of a chance
experiment.

Question 2

$$\text { For a discrete random variable, } \sigma _ { x } = \sum _ { x \text { values } } \left( x - \mu _ { x } \right) \cdot p ( x )$$

Accepted Answer

The correct formula for the standard deviation of a discrete random variable is $\sigma_x = \sqrt{\sum_{x \text{ values }} (x - \mu_x)^2 \cdot p(x)}$. The given expression is for the expected deviation, not the standard deviation.

Question 3

The standard normal distribution has a mean equal to 0 and standard
deviation equal to 1.

Accepted Answer

The statement is true.

Question 4

A random variable is discrete if its value depends upon the outcome of a
chance experiment.

Accepted Answer

The answer of A random variable is discrete if its...

Question 5

Using the notation C = continuous and D = discrete, indicate whether each of the
following random variables is discrete or continuous.
a) The number of defective lights in a school's main hallway on a randomly selected
day
b) The barometric pressure at midnight in a particular location in Iowa on a randomly
selected day
c) The number of staples left in a stapler at a randomly selected time
d) The number of sentences in a short story selected at random from a collection of
short stories
e) The oven temperature at a randomly selected time during the cooking of a turkey

Accepted Answer

The answer of Using the notation C = continuous and...

Question 6

A random variable is continuous if its possible values are all points in
some interval.

Accepted Answer

This statement defines a continuous random variable, which is true.

Question 7

What is a random variable?

Accepted Answer

The answer of What is a random variable?...

Question 8

$$\text { For random variables } x \text { and } y \text {, if } y = a + b x \text {, then } \sigma _ { y } = a + b \sigma _ { x }$$

Accepted Answer

The correct formula should be $\sigma_y = |b|\sigma_x$ because when the sign of $b$ changes, it can change the variability of $y$. Adding a constant term ($a$) does not affect the variability.

Question 9

At the College of Warm & Fuzzy, good grades in math are easy to come by. The
grade distribution is given in the table below: $$\begin{array} { | l | c | c | c | c | c | } 
\hline \text { Grade } & \text { A } & \text { B } & \text { C } & \text { D } & \text { F } \
\hline \text { Proportion } & 0.30 & 0.35 & 0.25 & 0.09 & 0.01 \
\hline
\end{array}$$ Suppose three students' grade reports are selected at random with replacement. (The
math grade is written down and the report replaced in the population of reports before
the next report is selected.) Three possible outcomes of this experiment are listed
below. Calculate the probabilities of these sequences appearing. a) $\mathbf { B A C }$
b) $\mathbf { A } \mathbf { A B }$

Accepted Answer

The answer of At the College of Warm & Fuzzy,...

Question 10

If $x$ is a random variable, and random variable $y$ is defined as $y = a + b x$, then $\mu _ { y } = | b | \sigma _ { x }$.

Accepted Answer

The mean of $y$, $\mu_y$, is given by $\mu_y = a + b\mu_x$, not $|b|\sigma_x$. The expression $|b|\sigma_x$ relates to the standard deviation, not the mean.

Question 11

The density curve for a continuous random variable is shown below. Use this curve
to find the following probabilities: 
a) $P ( x < 1 )$
b) $P ( x$ is at least $3 )$
You may need to use the following area formulas in your calculations:
Area of a rectangle: $A = l w$
Area of a trapezoid: $A = \frac { 1 } { 2 } h \left( b _ { 1 } + b _ { 2 } \right)$
Area of a right triangle: $A = \frac { 1 } { 2 } a b$

Accepted Answer

The answer of The density curve for a continuous random...

Question 12

The probability of $k$ successes among $n$ independent trials, each with equal probability of success, $p$, is: $\frac { n ! } { x ! ( n - x ) ! } p ^ { k } ( 1 - p ) ^ { n - k }$.

Accepted Answer

This is the formula for the Binomial distribution, which describes the probability of a certain number of successes in a fixed number of independent trials with a constant probability of success.

Question 13

For a continuous random variable x, the area under the density curve above
an interval a to b represents the probability that the value of x is between a
and b.

Accepted Answer

This is the definition of the probability density function for a continuous random variable. The probability of x lying between a and b is equal to the area under the density curve between a and b.

Question 14

For every random variable, both discrete and continuous,
$$P ( a \leq x \leq b ) = P ( a < x < b ) \text {. }$$

Accepted Answer

This statement is only true for continuous random variables. For discrete random variables, since the probability of any single point is zero, we need to use the less than or equal to operator (instead of just the less than operator) to capture the probability of the endpoints. Therefore, for discrete random variables, $P(a \leq x \leq b) 
eq P(a < x < b)$.

Question 15

Road construction in the town of Hiawatha has presented some problems for the
traffic engineers. Small businesses such as Bev's Hair Gallery will require an
alternate entrance for their customers. This will in turn create a minor congestion of
traffic nearby because cars will be backed up and the entrance to the parking lot. The
engineers have studied the problem using simulations based on current traffic
patterns. The results from 500 trials are shown below: $$\begin{array} { | c | c | } 
\hline \boldsymbol { k } & \begin{array} { l } 
\text { Number of times } \boldsymbol { k } \
\text { cars are backed up } \
\text { at the entrance }
\end{array} \
\hline 0 & 250 \
1 & 150 \
2 & 75 \
3 & 25 \
\hline
\end{array}$$ Let the random variable k = number of cars backed up at the entrance.
a) Fill in the table below with the estimated probability distribution of k, and sketch
a probability histogram for x. Probability distribution
$\begin{array}{|l|l|}
\hline k & \mathrm{P}(x) \
\hline \mathbf{0} & \
\hline \mathbf{1} & \
\hline 2 & \
\hline 3 & \
\hline
\end{array}$

Probability histogram

b) Using the estimated probabilities in part (a), calculate the following: i) $P ( k = 1 )$, the probability that 1 car is backed up at the entrance.
ii) $P ( x < 2 )$, the probability that fewer than 2 cars are backed up at the entrance
iii) $P ( x \geq 1 )$, the probability that at least 1 car is backed up at the entrance

Accepted Answer

The answer of Road construction in the town of Hiawatha...

Question 16

$$\text { If random variables } x _ { 1 } \text { and } x _ { 2 } \text { are independent, then } \sigma _ { x _ { 1 } + x _ { 2 } } ^ { 2 } = \sigma _ { x _ { 1 } } ^ { 2 } + \sigma _ { x _ { 2 } } ^ { 2 }$$

Accepted Answer

If two random variables, $x_1$ and $x_2$, are independent, their variances add up. That is, $\operatorname{Var}(x_1+x_2) = \operatorname{Var}(x_1)+\operatorname{Var}(x_2)$. Taking the square root of both sides gives the desired result. That is, $\sigma_{x_1+x_2} = \sqrt{\operatorname{Var}(x_1+x_2)} = \sqrt{\operatorname{Var}(x_1)+\operatorname{Var}(x_2)} = \sqrt{\sigma_{x_1}^2+\sigma_{x_2}^2}$.

Question 17

$$\text { For a discrete random variable } x , \mu _ { x } = \sum _ { x \text { values } } x \cdot p ( x )$$

Accepted Answer

This is the formula for the expected value (also known as the mean) of a discrete random variable.

Question 18

A normal probability plot suggests that a normal probability model is
plausible if there is no obvious pattern in the scatter of points.

Accepted Answer

The answer of A normal probability plot suggests that a...

Question 19

The probability that in a sequence of independent trials, each with equal probability of success, $p$, the first success at trial $x$ is: $p ^ { x - 1 } ( 1 - p )$.

Accepted Answer

The correct probability is $ (1 - p)^{x - 1} p $, where $ (1 - p)^{x - 1} $ represents the probability of $x-1$ failures followed by a success.

Question 20

The distribution of all values of a random variable is called a normal
distribution.

Accepted Answer

The distribution of all values of a random variable may take on many different forms, one of which is the normal distribution.

Quiz 6: Random Variables and Probability Distributions

What is a random variable?

$\text { For a discrete random variable, } \sigma _ { x } = \sum _ { x \text { values } } \left( x - \mu _ { x } \right) \cdot p ( x )$

The standard normal distribution has a mean equal to 0 and standard deviation equal to 1.

A random variable is discrete if its value depends upon the outcome of a chance experiment.

A random variable is continuous if its possible values are all points in some interval.

What is a random variable?

$\text { For random variables } x \text { and } y \text {, if } y = a + b x \text {, then } \sigma _ { y } = a + b \sigma _ { x }$

If $x$ is a random variable, and random variable $y$ is defined as $y = a + b x$ , then $\mu _ { y } = | b | \sigma _ { x }$ .

The probability of $k$ successes among $n$ independent trials, each with equal probability of success, $p$ , is: $\frac { n ! } { x ! ( n - x ) ! } p ^ { k } ( 1 - p ) ^ { n - k }$ .

For a continuous random variable x, the area under the density curve above an interval a to b represents the probability that the value of x is between a and b.

For every random variable, both discrete and continuous, $P ( a \leq x \leq b ) = P ( a < x < b ) \text {. }$

$\text { If random variables } x _ { 1 } \text { and } x _ { 2 } \text { are independent, then } \sigma _ { x _ { 1 } + x _ { 2 } } ^ { 2 } = \sigma _ { x _ { 1 } } ^ { 2 } + \sigma _ { x _ { 2 } } ^ { 2 }$

$\text { For a discrete random variable } x , \mu _ { x } = \sum _ { x \text { values } } x \cdot p ( x )$

A normal probability plot suggests that a normal probability model is plausible if there is no obvious pattern in the scatter of points.

The probability that in a sequence of independent trials, each with equal probability of success, $p$ , the first success at trial $x$ is: $p ^ { x - 1 } ( 1 - p )$ .

The distribution of all values of a random variable is called a normal distribution.

Quiz 6: Random Variables and Probability Distributions

What is a random variable?

For a discrete random variable, σx=∑x values (x−μx)⋅p(x)\text { For a discrete random variable, } \sigma _ { x } = \sum _ { x \text { values } } \left( x - \mu _ { x } \right) \cdot p ( x ) For a discrete random variable, σx​=∑x values ​(x−μx​)⋅p(x)

The standard normal distribution has a mean equal to 0 and standard deviation equal to 1.

A random variable is discrete if its value depends upon the outcome of a chance experiment.

A random variable is continuous if its possible values are all points in some interval.

What is a random variable?

For random variables x and y, if y=a+bx, then σy=a+bσx\text { For random variables } x \text { and } y \text {, if } y = a + b x \text {, then } \sigma _ { y } = a + b \sigma _ { x } For random variables x and y, if y=a+bx, then σy​=a+bσx​

If xxx is a random variable, and random variable yyy is defined as y=a+bxy = a + b xy=a+bx , then μy=∣b∣σx\mu _ { y } = | b | \sigma _ { x }μy​=∣b∣σx​ .

The probability of kkk successes among nnn independent trials, each with equal probability of success, ppp , is: n!x!(n−x)!pk(1−p)n−k\frac { n ! } { x ! ( n - x ) ! } p ^ { k } ( 1 - p ) ^ { n - k }x!(n−x)!n!​pk(1−p)n−k .

For a continuous random variable x, the area under the density curve above an interval a to b represents the probability that the value of x is between a and b.

For every random variable, both discrete and continuous, P(a≤x≤b)=P(a<x<b). P ( a \leq x \leq b ) = P ( a < x < b ) \text {. }P(a≤x≤b)=P(a<x<b).

For a discrete random variable x,μx=∑x values x⋅p(x)\text { For a discrete random variable } x , \mu _ { x } = \sum _ { x \text { values } } x \cdot p ( x ) For a discrete random variable x,μx​=∑x values ​x⋅p(x)

A normal probability plot suggests that a normal probability model is plausible if there is no obvious pattern in the scatter of points.

The probability that in a sequence of independent trials, each with equal probability of success, ppp , the first success at trial xxx is: px−1(1−p)p ^ { x - 1 } ( 1 - p )px−1(1−p) .

The distribution of all values of a random variable is called a normal distribution.

$\text { For a discrete random variable, } \sigma _ { x } = \sum _ { x \text { values } } \left( x - \mu _ { x } \right) \cdot p ( x )$

$\text { For random variables } x \text { and } y \text {, if } y = a + b x \text {, then } \sigma _ { y } = a + b \sigma _ { x }$

If $x$ is a random variable, and random variable $y$ is defined as $y = a + b x$ , then $\mu _ { y } = | b | \sigma _ { x }$ .

The probability of $k$ successes among $n$ independent trials, each with equal probability of success, $p$ , is: $\frac { n ! } { x ! ( n - x ) ! } p ^ { k } ( 1 - p ) ^ { n - k }$ .

For every random variable, both discrete and continuous, $P ( a \leq x \leq b ) = P ( a < x < b ) \text {. }$

$\text { For a discrete random variable } x , \mu _ { x } = \sum _ { x \text { values } } x \cdot p ( x )$

The probability that in a sequence of independent trials, each with equal probability of success, $p$ , the first success at trial $x$ is: $p ^ { x - 1 } ( 1 - p )$ .