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?

<p>$Var(X) = ∑ (𝑋 - µ)^2 P(x)$ (B)</p>
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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 (B)</p>
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What is the purpose of finding the sample mean in statistics?

<p>To estimate the population mean (C)</p>
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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 (D)</p>
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What is the formula for calculating the variance of a discrete random variable $X$ with probabilities $p(x)$?

<p>$\sigma^2 = \sum (x - \mu)^2 p(x)$ (D)</p>
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If the mean of a discrete random variable is $\mu = 5$ and the variance is $\sigma^2 = 4$, what is the standard deviation?

<p>$\sigma = 2$ (A)</p>
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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>$\sigma^2 = 1.8$ (B)</p>
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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>$s^2 = 9$ (B)</p>
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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 (B)</p>
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What is the formula used to calculate the variance of the probability distribution?

<p>$\sigma^2 = \sum (x - \mu)^2 p(x)$ (B)</p>
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What is the value of the standard deviation calculated from the given probability distribution?

<p>1.25 (D)</p>
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What is the mean of the probability distribution for the number of heads when three coins are tossed?

<p>1.5 (C)</p>
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Which of the following is the correct expression for the variance of the probability distribution?

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