Question 1

Find the expected value of a random variable X having the following probability distribution: $$\begin{array} { c c c c c } 
x & 2 & 4 & 6 & 8 \
P ( X = x ) & \frac { 7 } { 20 } & \frac { 2 } { 20 } & \frac { 5 } { 20 } & \frac { 6 } { 20 }
\end{array}$$ ​ $$E ( X ) =$$ __________

Accepted Answer

The answer of Find the expected value of a random...

Question 2

Calculate the expected value of X for the given probability distribution. $$\begin{array} { c c c c c } 
x & 0 & 1 & 2 & 3 \
P ( X = x ) & 0.4 & 0.2 & 0.2 & 0.2
\end{array}$$ ​ $$E ( X ) =$$ __________

Accepted Answer

The answer of Calculate the expected value of X for...

Question 3

Calculate the expected value of X for the given probability distribution: $$\begin{array} { c c c c c c c } 
x & - 5 & - 1 & 0 & 2 & 5 & 10 \
P ( X = x ) & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.5
\end{array}$$   Round your answer to two decimal places if necessary.

Accepted Answer

Expected value is the sum of the product of each value and its probability. So, $E ( X ) = - 5 ( 0.1 ) + ( - 1 ) ( 0.1 ) + 0 ( 0.1 ) + 2 ( 0.1 ) + 5 ( 0.1 ) + 10 ( 0.5 ) = 5.1$

Question 4

Select five cards without replacement from a standard deck of 52, and let X be the number of black cards you draw. Calculate the expected value of the given random variable X.

Accepted Answer

The expected value of drawing a black card from a standard deck of 52 cards is calculated by multiplying the probability of drawing a black card by the total number of draws. Since half of the cards (26 out of 52) are black, the probability of drawing a black card in any single draw is 26/52 = 1/2. When drawing five cards, the expected number of black cards is 5 * (1/2) = 2.5.

Question 5

The following table shows the approximate numbers of school goers in the U.S. (residents who attended some educational institution) in 1998, broken down by age group. Use the rounded midpoints of the given measurement classes to compute the probability distribution of the age  X of a school goer. (Round probabilities to two decimal places.) Hence compute the expected value of X. $$\begin{array} { c c c c c c c } 
\text { Age } & 3 - 6.9 & 7 - 12.9 & 13 - 16.9 & 17 - 22.9 & 23 - 26.9 & 27 - 42.9 \
\begin{array} { c } 
\text { Population } \
\text { (Millions) }
\end{array} & 11 & 22 & 16 & 15 & 4 & 7
\end{array}$$

Accepted Answer

To calculate the expected value $E(X)$ of the age $X$ of a school goer, we first need to find the probability distribution of $X$, and then use these probabilities to compute the expected value. The midpoint for each age group is calculated as the average of the lower and upper bounds of the age group. Then, we find the probability of each age group by dividing the population of the group by the total population. Finally, we multiply each midpoint by its corresponding probability and sum these products to get the expected value.1. Calculate midpoints for each age group:- $3 - 6.9$: midpoint = $\frac{3 + 6.9}{2} = 4.95$- $7 - 12.9$: midpoint = $\frac{7 + 12.9}{2} = 9.95$- $13 - 16.9$: midpoint = $\frac{13 + 16.9}{2} = 14.95$- $17 - 22.9$: midpoint = $\frac{17 + 22.9}{2} = 19.95$- $23 - 26.9$: midpoint = $\frac{23 + 26.9}{2} = 24.95$- $27 - 42.9$: midpoint = $\frac{27 + 42.9}{2} = 34.95$2. Calculate the total population:Total population = $11 + 22 + 16 + 15 + 4 + 7 = 75$ million3. Calculate the probability of each age group:- $3 - 6.9$: $P = \frac{11}{75}$- $7 - 12.9$: $P = \frac{22}{75}$- $13 - 16.9$: $P = \frac{16}{75}$- $17 - 22.9$: $P = \frac{15}{75}$- $23 - 26.9$: $P = \frac{4}{75}$- $27 - 42.9$: $P = \frac{7}{75}$4. Calculate the expected value $E(X)$:$$E(X) = 4.95\left(\frac{11}{75}\right) + 9.95\left(\frac{22}{75}\right) + 14.95\left(\frac{16}{75}\right) + 19.95\left(\frac{15}{75}\right) + 24.95\left(\frac{4}{75}\right) + 34.95\left(\frac{7}{75}\right)$$After performing the calculations and rounding to two decimal places, the expected value $E(X)$ is approximately 15.2.

Question 6

Calculate the expected value of X for the given probability distribution. $$\begin{array} { c c c c c } 
x & 1 & 2 & 3 & 4 \
P ( X = x ) & 0.4 & 0.2 & 0.2 & 0.2
\end{array}$$ ​ $$E ( X ) =$$ __________

Accepted Answer

The answer of Calculate the expected value of X for...

Question 7

Find the expected value of a random variable X having the following probability distribution: $$\begin{array} { c c c c c } 
x & 10 & 20 & 30 & 40 \
P ( X = x ) & \frac { 25 } { 60 } & \frac { 20 } { 60 } & \frac { 5 } { 60 } & \frac { 10 } { 60 }
\end{array}$$ ​ $$E ( X ) =$$ __________

Accepted Answer

The answer of Find the expected value of a random...

Question 8

Given the probability distribution below, calculate the expected value of X. $$\begin{array} { c c c c c c c } 
x & - 4 & - 2 & 0 & 3 & 7 & 16 \
P ( X = x ) & 0.1 & 0.2 & 0.3 & 0.2 & 0 & 0.2
\end{array}$$ ​ $$\mu = E ( X )$$ __________​

Accepted Answer

The answer of Given the probability distribution below, calculate the...

Question 9

The following is a sample of tow ratings (in pounds) for some popular 2000 model sport utility vehicles.  
6,400, 12,800, 19,200, 25,600, 32,000, 38,400
 
Compute the median.

Accepted Answer

The median is the middle value of a data set when it is ordered. For an even number of observations, it is the average of the two middle numbers. Here, the middle values are 19,200 and 25,600, so the median is (19,200 + 25,600) / 2 = 22,400.

Question 10

Given the data.   $2.4 , - 5.2,4.7 , - 0.5 , - 0.5$  
Compute the median.

Accepted Answer

To find the median, first arrange the data in ascending order: $-5.2, -0.5, -0.5, 2.4, 4.7$. The median is the middle number, which is $-0.5$ when the data is ordered.

Question 11

The following figures show the price of silver in dollars per ounce for the 10-business day period April 4 - April 15, 2004. ​
7)0, 6.7, 7.0, 7.3, 7.0, 7.4, 7.4, 7.4, 7.3, 7.3
​
Find the sample median.
​

Accepted Answer

To find the median, we need to arrange the numbers in ascending or descending order:

6.7, 7.0, 7.0, 7.3, 7.3, 7.3, 7.4, 7.4, 7.4, 7.4

Since there are 10 values, the median is the average of the fifth and sixth values, which is (7.3+7.3)/2 = 7.3. Therefore, the answer is A.

Question 12

Let X be the lower number when two dice are rolled. Calculate the expected value of the given random variable X. Please, round your answer to the nearest thousandth.

Accepted Answer

The expected value of the lower number (X) when two dice are rolled can be calculated by considering all the possible outcomes and their probabilities. For each possible lower number (1 through 6), we calculate its probability by counting how many outcomes result in that number being the lower of the two and then multiply each by the value of the lower number, summing these products to get the expected value. The correct calculation leads to an expected value of approximately 2.528.

Question 13

A roulette wheel has the numbers 1 through 36, 0, and 00. A bet on two numbers pays 17 to 1 (that is, if one of the two numbers you bet comes up, you get back your $1 plus another $17). How much do you expect to win with a $10 bet on two numbers  Please, round your answer to the nearest thousandth of a dollar. ?

Accepted Answer

The expected value calculation for a $10 bet on two numbers in roulette, considering the 38 possible outcomes (1-36, 0, 00), is: [(2/38) * $170] - [(36/38) * $10] = -$0.5263, rounded to -$0.53, indicating an expected loss of $0.53 per spin.

Question 14

Give a sample of 8 scores with mean 1 and with median ≠ mean. Arrange the scores in ascending order. ​

Accepted Answer

The mean of the scores in option B is 1, calculated as (0+0+0+0+0+0+0+8)/8 = 8/8 = 1. The median, which is the average of the 4th and 5th scores (both 0), is not equal to the mean.

Question 15

Let X be the number of tails that come up when a coin is tossed three times. Calculate the expected value of the given random variable X.

Accepted Answer

The expected value $E(X)$ of a random variable $X$ representing the number of tails in three coin tosses is calculated by multiplying each outcome by its probability and summing these products. The probability of getting a tail in a single toss is $0.5$. For three tosses, the expected value is $3 \times 0.5 = 1.5$.

Question 16

The following table shows the distribution of household incomes for a sample of 1,000 households in the U.S. with incomes up to $100,000.

Use this information to estimate, to the nearest $1,000, the average household income for such households.

Accepted Answer

The answer of The following table shows the distribution of...

Question 17

12 darts are thrown at a dartboard. The probability of hitting a bull's-eye is 0.1. Let X be the number of bull's-eyes hit. Calculate the expected value of the given random variable X.

Accepted Answer

The expected value $E(X)$ for a binomial distribution is calculated as $n \times p$, where $n$ is the number of trials (12 darts) and $p$ is the probability of success (0.1). Thus, $E(X) = 12 \times 0.1 = 1.2$.

Question 18

The following is a sample of the percentage increases in the price of a house from 1980 to 2001 in 8 regions of the U.S.   $75,120,145,155,155,225,225,300$  
Compute the median of the given sample.

Accepted Answer

To find the median, first arrange the data in ascending order: $75, 120, 145, 155, 155, 225, 225, 300$. Since there are 8 numbers, the median is the average of the 4th and 5th numbers, which are both 155. Therefore, the median $m = 155$.

Question 19

Given the data.   $4,0 , - 4,4,6$  
Compute the mean.

Accepted Answer

The mean $\bar{x}$ is calculated by summing all the numbers and dividing by the count of numbers. Here, $4 + 0 + (-4) + 4 + 6 = 10$, and there are 5 numbers, so $10 / 5 = 2$.

Question 20

Calculate the expected value of X for the given probability distribution: $$\begin{array} { c c c c c c c } 
x & - 20 & - 10 & 0 & 10 & 20 & 30 \
P ( X = x ) & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.5
\end{array}$$   Round your answer to two decimal places if necessary.

Accepted Answer

The expected value $E(X)$ is calculated by summing the products of each value of $x$ and its corresponding probability $P(X=x)$: $$E(X) = (-20)(0.1) + (-10)(0.1) + (0)(0.1) + (10)(0.1) + (20)(0.1) + (30)(0.5) = -2 - 1 + 0 + 1 + 2 + 15 = 15.$$

Quiz 9: Random Variables and Statistics

Find the expected value of a random variable X having the following probability distribution: $\begin{array} { c c c c c } x & 2 & 4 & 6 & 8 \\P ( X = x ) & \frac { 7 } { 20 } & \frac { 2 } { 20 } & \frac { 5 } { 20 } & \frac { 6 } { 20 }\end{array}$ $E ( X ) =$ __________

Calculate the expected value of X for the given probability distribution. $\begin{array} { c c c c c } x & 0 & 1 & 2 & 3 \\P ( X = x ) & 0.4 & 0.2 & 0.2 & 0.2\end{array}$ $E ( X ) =$ __________

Calculate the expected value of X for the given probability distribution: $\begin{array} { c c c c c c c } x & - 5 & - 1 & 0 & 2 & 5 & 10 \\P ( X = x ) & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.5\end{array}$ Round your answer to two decimal places if necessary.

Select five cards without replacement from a standard deck of 52, and let X be the number of black cards you draw. Calculate the expected value of the given random variable X.

Calculate the expected value of X for the given probability distribution. $\begin{array} { c c c c c } x & 1 & 2 & 3 & 4 \\P ( X = x ) & 0.4 & 0.2 & 0.2 & 0.2\end{array}$ $E ( X ) =$ __________

Find the expected value of a random variable X having the following probability distribution: $\begin{array} { c c c c c } x & 10 & 20 & 30 & 40 \\P ( X = x ) & \frac { 25 } { 60 } & \frac { 20 } { 60 } & \frac { 5 } { 60 } & \frac { 10 } { 60 }\end{array}$ $E ( X ) =$ __________

Given the probability distribution below, calculate the expected value of X. $\begin{array} { c c c c c c c } x & - 4 & - 2 & 0 & 3 & 7 & 16 \\P ( X = x ) & 0.1 & 0.2 & 0.3 & 0.2 & 0 & 0.2\end{array}$ $\mu = E ( X )$ __________

The following is a sample of tow ratings (in pounds) for some popular 2000 model sport utility vehicles. 6,400, 12,800, 19,200, 25,600, 32,000, 38,400 Compute the median.

Given the data. $2.4 , - 5.2,4.7 , - 0.5 , - 0.5$ Compute the median.

The following figures show the price of silver in dollars per ounce for the 10-business day period April 4 - April 15, 2004. 7) 0, 6.7, 7.0, 7.3, 7.0, 7.4, 7.4, 7.4, 7.3, 7.3 Find the sample median.

Let X be the lower number when two dice are rolled. Calculate the expected value of the given random variable X. Please, round your answer to the nearest thousandth.

Give a sample of 8 scores with mean 1 and with median ≠ mean. Arrange the scores in ascending order.

Let X be the number of tails that come up when a coin is tossed three times. Calculate the expected value of the given random variable X.

12 darts are thrown at a dartboard. The probability of hitting a bull's-eye is 0.1. Let X be the number of bull's-eyes hit. Calculate the expected value of the given random variable X.

The following is a sample of the percentage increases in the price of a house from 1980 to 2001 in 8 regions of the U.S. $75,120,145,155,155,225,225,300$ Compute the median of the given sample.

Given the data. $4,0 , - 4,4,6$ Compute the mean.

Calculate the expected value of X for the given probability distribution: $\begin{array} { c c c c c c c } x & - 20 & - 10 & 0 & 10 & 20 & 30 \\P ( X = x ) & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.5\end{array}$ Round your answer to two decimal places if necessary.

Quiz 9: Random Variables and Statistics

Calculate the expected value of X for the given probability distribution. x0123P(X=x)0.40.20.20.2\begin{array} { c c c c c } x & 0 & 1 & 2 & 3 \\P ( X = x ) & 0.4 & 0.2 & 0.2 & 0.2\end{array}xP(X=x)​00.4​10.2​20.2​30.2​ ​ E(X)=E ( X ) =E(X)= __________

Select five cards without replacement from a standard deck of 52, and let X be the number of black cards you draw. Calculate the expected value of the given random variable X.

Calculate the expected value of X for the given probability distribution. x1234P(X=x)0.40.20.20.2\begin{array} { c c c c c } x & 1 & 2 & 3 & 4 \\P ( X = x ) & 0.4 & 0.2 & 0.2 & 0.2\end{array}xP(X=x)​10.4​20.2​30.2​40.2​ ​ E(X)=E ( X ) =E(X)= __________

The following is a sample of tow ratings (in pounds) for some popular 2000 model sport utility vehicles. 6,400, 12,800, 19,200, 25,600, 32,000, 38,400 Compute the median.

Given the data. 2.4,−5.2,4.7,−0.5,−0.52.4 , - 5.2,4.7 , - 0.5 , - 0.52.4,−5.2,4.7,−0.5,−0.5 Compute the median.

The following figures show the price of silver in dollars per ounce for the 10-business day period April 4 - April 15, 2004. ​ 7) 0, 6.7, 7.0, 7.3, 7.0, 7.4, 7.4, 7.4, 7.3, 7.3 ​ Find the sample median. ​

Let X be the lower number when two dice are rolled. Calculate the expected value of the given random variable X. Please, round your answer to the nearest thousandth.

Give a sample of 8 scores with mean 1 and with median ≠ mean. Arrange the scores in ascending order. ​

Let X be the number of tails that come up when a coin is tossed three times. Calculate the expected value of the given random variable X.

The following table shows the distribution of household incomes for a sample of 1,000 households in the U.S. with incomes up to $100,000. Use this information to estimate, to the nearest $1,000, the average household income for such households.

12 darts are thrown at a dartboard. The probability of hitting a bull's-eye is 0.1. Let X be the number of bull's-eyes hit. Calculate the expected value of the given random variable X.

The following is a sample of the percentage increases in the price of a house from 1980 to 2001 in 8 regions of the U.S. 75,120,145,155,155,225,225,30075,120,145,155,155,225,225,30075,120,145,155,155,225,225,300 Compute the median of the given sample.

Given the data. 4,0,−4,4,64,0 , - 4,4,64,0,−4,4,6 Compute the mean.

Calculate the expected value of X for the given probability distribution. $\begin{array} { c c c c c } x & 0 & 1 & 2 & 3 \\P ( X = x ) & 0.4 & 0.2 & 0.2 & 0.2\end{array}$ $E ( X ) =$ __________

Calculate the expected value of X for the given probability distribution. $\begin{array} { c c c c c } x & 1 & 2 & 3 & 4 \\P ( X = x ) & 0.4 & 0.2 & 0.2 & 0.2\end{array}$ $E ( X ) =$ __________

Given the data. $2.4 , - 5.2,4.7 , - 0.5 , - 0.5$ Compute the median.

The following figures show the price of silver in dollars per ounce for the 10-business day period April 4 - April 15, 2004. 7) 0, 6.7, 7.0, 7.3, 7.0, 7.4, 7.4, 7.4, 7.3, 7.3 Find the sample median.

Give a sample of 8 scores with mean 1 and with median ≠ mean. Arrange the scores in ascending order.

The following is a sample of the percentage increases in the price of a house from 1980 to 2001 in 8 regions of the U.S. $75,120,145,155,155,225,225,300$ Compute the median of the given sample.

Given the data. $4,0 , - 4,4,6$ Compute the mean.