Statistics and Probability Lesson 3: Mean, Variance, SD of Discrete Probability Distribution

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Questions and Answers

What does the mean of a discrete random variable represent?

The spread of the distribution

The probability of each value

The sum of all values

The weighted average of possible values (correct)

In finding the mean of a discrete probability distribution, what is the first step?

Calculate the sum of X * P(X)

Subtract the computed mean from each value

Construct the probability distribution (correct)

Determine the value of X * P(X)

What does the variance of a discrete random variable measure?

The spread of the distribution

The probability of the random variable

The variability of the distribution (correct)

The sum of all values

Which step is involved in finding the variance of a discrete probability distribution?

Multiply each (X - mean)² by its corresponding probability Signup and view all the answers

What does ∑(X - 𝝁)² * P(X) represent in the variance formula of a discrete probability distribution?

The sum of squared differences between values and mean Signup and view all the answers

When finding the mean, what is multiplied by the corresponding probabilities?

(X - mean) Signup and view all the answers

What is the formula for calculating the standard deviation of a probability distribution?

√∑(𝑿 − 𝝁)𝟐 ∙ 𝑷(𝑿) Signup and view all the answers

What is the purpose of Step 3 in finding the mean, variance, and standard deviation of a probability distribution?

Finding the mean of the expected value Signup and view all the answers

In Step 6 of calculating the probability distribution, what operation is performed on each result obtained in Step 5?

Multiplying Signup and view all the answers

What is Step 8 primarily responsible for in the process of finding the standard deviation?

Finding the square root of the variance Signup and view all the answers

If 𝝈 = 0.866, what does this value represent in relation to the probability distribution?

Standard Deviation Signup and view all the answers

What does the standard deviation of a random variable measure?

The average distance from the mean Signup and view all the answers

In the context of a discrete probability distribution, what does 𝜇 represent?

Mean Signup and view all the answers

How is the variance calculated for a discrete random variable in this context?

By multiplying each value by its corresponding probability and summing the results Signup and view all the answers

What is the formula to calculate the standard deviation for a discrete probability distribution?

∑(𝑿 − 𝝁)𝟐 ∙ 𝑷(𝑿) Signup and view all the answers

What does Step 7 in the process entail when calculating the variance?

Summing the results obtained in Step 6 Signup and view all the answers

How can the standard deviation be obtained from the variance value?

Take the square root of the variance Signup and view all the answers

Study Notes

Mean of a Discrete Probability Distribution

The mean of a discrete random variable X is a weighted average of the possible values that the random variable can take.
Formula: 𝝁 = ∑ 𝑿 ∙ 𝑷(𝑿)
Steps to find the mean:
- Construct the probability distribution for the random variable.
- Determine the value of 𝑿 ∙ 𝑷(𝑿)
- Add all the values of to 𝑿 ∙ 𝑷(𝑿) to determine ∑ 𝑿 ∙ 𝑷(𝑿)

Variance of a Discrete Probability Distribution

The variance of a discrete random variable X measures the spread, or variability, of the distribution.
Formula: 𝝈𝟐 = ∑(𝑿 − 𝝁)𝟐 ∙ 𝑷(𝑿)
Steps to find the variance:
- Subtract the computed mean from each value of the random variable: 𝑿 − 𝝁
- Square the value obtained: (𝑿 − 𝝁)𝟐
- Multiply the value obtained by the given probability: (𝑿 − 𝝁)𝟐 ∙ 𝑷(𝑿)
- Find the variance by adding all products obtained

Standard Deviation of a Discrete Probability Distribution

The standard deviation of a random variable measures how close the random variable is to the mean.
Formula: 𝝈 = √∑(𝑿 − 𝝁)𝟐 ∙ 𝑷(𝑿)
Steps to find the standard deviation:
- Find the variance: 𝝈𝟐 = ∑(𝑿 − 𝝁)𝟐 ∙ 𝑷(𝑿)
- Take the square root of the variance: 𝝈 = √𝝈𝟐

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Description

Learn how to compute the mean, variance, and standard deviation of a discrete probability distribution. Understand the formula for calculating the mean of a discrete random variable and the steps involved in finding it.