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# Lesson 2: Variance and Standard Deviation of Discrete Random Variables

Learn how to calculate the spread or dispersion in a probability distribution by computing the variance and standard deviation of a discrete probability distribution. This lesson covers understanding and applying these key statistical measures.

Created by
@LivelyBongos

## Questions and Answers

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

$µ = ∑ 𝑋𝑃(𝑥)$

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

To find the difference between the values and the mean

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

Both steps 4 and 5

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

<p>$Var(X) = ∑ (𝑋 - µ)^2 P(x)$</p> Signup and view all the answers

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

<p>The standard deviation is the square root of the variance</p> Signup and view all the answers

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

<p>To estimate the population mean</p> Signup and view all the answers

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

<p>To measure the amount of spread or dispersion in the distribution</p> Signup and view all the answers

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

<p>$\sigma^2 = \sum (x - \mu)^2 p(x)$</p> 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?

<p>$\sigma = 2$</p> 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?

<p>$\sigma^2 = 1.8$</p> 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?

<p>$s^2 = 9$</p> Signup and view all the answers

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

<p>The standard deviation is the square root of the variance</p> Signup and view all the answers

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

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

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

<p>1.25</p> Signup and view all the answers

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

<p>1.5</p> Signup and view all the answers

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

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

### What is the purpose of finding the variance and standard deviation of a probability distribution?

<p>To measure the spread or dispersion of the distribution</p> Signup and view all the answers

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