Probability Distribution, Mean, and Standard Deviation for a Discrete Random Variable

Q1. Does the table show a probability distribution?

Background

Topic: Probability Distributions

This question tests your understanding of the requirements for a probability distribution for a discrete random variable. You need to check if the values of $x$ are associated with valid probabilities and if those probabilities meet certain criteria.

Key Terms:

• Discrete random variable: A variable that can take on a countable number of values.

• Probability distribution: A table or function that gives the probabilities of each possible value of a random variable.

Requirements for a Probability Distribution:

• Each probability $P(x)$ must be between 0 and 1 inclusive.

• The sum of all probabilities must equal 1: $\sum P(x) = 1$.

• The random variable $x$ must be numerical.

Step-by-Step Guidance

1. Check that all $P(x)$ values are between 0 and 1. Look at the table and verify each probability.

2. Sum all the probabilities: $0.029 + 0.142 + 0.334 + 0.358 + 0.137$. Is the total equal to 1?

3. Confirm that $x$ is a numerical random variable (number of households with a printer).

Try solving on your own before revealing the answer!

Q2. Find the mean of the random variable $x$.

Background

Topic: Expected Value (Mean) of a Discrete Probability Distribution

This question is testing your ability to calculate the mean (expected value) of a discrete random variable using its probability distribution.

Key Formula:

$\mu = \sum [x \cdot P(x)]$

Where:

• $\mu$ = mean (expected value)

• $x$ = value of the random variable

• $P(x)$ = probability of $x$

Step-by-Step Guidance

1. Multiply each value of $x$ by its corresponding probability $P(x)$: $x \cdot P(x)$ for each row.

2. Add up all these products to get the mean: $\mu = \sum [x \cdot P(x)]$.

Try solving on your own before revealing the answer!

Q3. Find the standard deviation of the random variable $x$.

Background

Topic: Standard Deviation of a Discrete Probability Distribution

This question is testing your ability to calculate the standard deviation, which measures the spread of the random variable values around the mean.

Key Formula:

$\sigma = \sqrt{\sum [(x - \mu)^2 \cdot P(x)]}$

Where:

• $\sigma$ = standard deviation

• $x$ = value of the random variable

• $\mu$ = mean (expected value)

• $P(x)$ = probability of $x$

Step-by-Step Guidance

1. For each value of $x$, subtract the mean $\mu$ and square the result: $(x - \mu)^2$.

2. Multiply each squared difference by its probability: $(x - \mu)^2 \cdot P(x)$.

3. Add up all these products.

4. Take the square root of the sum to get the standard deviation: $\sigma = \sqrt{\sum [(x - \mu)^2 \cdot P(x)]}$.

Try solving on your own before revealing the answer!

Final Answers:

Q1: Yes, the table shows a probability distribution. All probabilities are between 0 and 1, the sum is 1, and $x$ is numerical.

Q2: $\mu = 2.4$ household(s). This is the expected number of households with a printer in a group of four.

Q3: $\sigma = 0.9$ household(s). This measures the variability in the number of households with a printer.