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Lesson 2: Variance and Standard Deviation of Discrete Random Variables

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Questions and Answers

What is the formula used to calculate the mean of the probability distribution?

$µ = ∑ 𝑋/𝑛$
$µ = ∑ 𝑋𝑃(𝑥)/𝑛$
$µ = ∑ 𝑋𝑃(𝑥)$ (correct)
$µ = ∑ 𝑋^2𝑃(𝑥)$

What is the purpose of subtracting the mean from each value of the random variable $X$ in step 2?

To calculate the variance
To center the data around zero
To calculate the standard deviation
To find the difference between the values and the mean (correct)

Which step is used to calculate the variance of the probability distribution?

Step 4, where the results from step 3 are multiplied by the corresponding probability
Step 5, where the sum of the results from step 4 is calculated
Both steps 4 and 5 (correct)
None of the above

What is the formula for calculating the variance of a probability distribution?

$Var(X) = ∑ (𝑋 - µ)^2 P(x)$ (B) Signup and view all the answers

What is the relationship between the variance and the standard deviation of a probability distribution?

The standard deviation is the square root of the variance (B) Signup and view all the answers

What is the purpose of finding the sample mean in statistics?

To estimate the population mean (C) Signup and view all the answers

What is the purpose of calculating the variance and standard deviation of a discrete probability distribution?

To measure the amount of spread or dispersion in the distribution (D) Signup and view all the answers

What is the formula for calculating the variance of a discrete random variable $X$ with probabilities $p(x)$?

$\sigma^2 = \sum (x - \mu)^2 p(x)$ (D) Signup and view all the answers

If the mean of a discrete random variable is $\mu = 5$ and the variance is $\sigma^2 = 4$, what is the standard deviation?

$\sigma = 2$ (A) Signup and view all the answers

The number of cars sold per day at a local car dealership has the following probability distribution:Number of Cars Sold (X): 0, 1, 2, 3, 4Probability (P(x)): 0.1, 0.2, 0.3, 0.2, 0.2What is the variance of this probability distribution?

$\sigma^2 = 1.8$ (B) Signup and view all the answers

If the sample mean of a set of data is $\bar{x} = 10$ and the sample standard deviation is $s = 3$, what is the sample variance?

$s^2 = 9$ (B) Signup and view all the answers

What is the relationship between the variance and standard deviation of a random variable?

The standard deviation is the square root of the variance (B) Signup and view all the answers

What is the formula used to calculate the variance of the probability distribution?

$\sigma^2 = \sum (x - \mu)^2 p(x)$ (B) Signup and view all the answers

What is the value of the standard deviation calculated from the given probability distribution?

1.25 (D) Signup and view all the answers

What is the mean of the probability distribution for the number of heads when three coins are tossed?

1.5 (C) Signup and view all the answers

Which of the following is the correct expression for the variance of the probability distribution?