Questions: Use the following probability distribution to answer the following questions. x P(x) ------ 3 0.37 6 0.1 7 0.08 9 0.13 10 0.07 15 0.11 20 0.14 Calculate the mean and standard deviation of the distribution. You may round your answers to two decimal places, if necessary. μ=□ σ=□ What is the expected value of the distribution? □

Use the following probability distribution to answer the following questions. x P(x) ------ 3 0.37 6 0.1 7 0.08 9 0.13 10 0.07 15 0.11 20 0.14 Calculate the mean and standard deviation of the distribution. You may round your answers to two decimal places, if necessary. μ=□ σ=□ What is the expected value of the distribution? □
Transcript text: Di2i Quizzes - 24FA_F2 MAT120 Probability and Statistics (W20) - Trident T... Module 2 Exam 100 points possible Answered:1/21 Question 2 Use the following probability distribution to answer the following questions. \begin{tabular}{|c|c|} \hline$x$ & $P(x)$ \\ \hline 3 & 0.37 \\ \hline 6 & 0.1 \\ \hline 7 & 0.08 \\ \hline 9 & 0.13 \\ \hline 10 & 0.07 \\ \hline 15 & 0.11 \\ \hline 20 & 0.14 \\ \hline \end{tabular} Calculate the mean and standard deviation of the distribution. You may round your answers to two decimal places, if necessary. \[ \begin{array}{l} \mu=\square \\ \sigma=\square \end{array} \] What is the expected value of the distribution? $\square$ MacBook Next Question
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Solution

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Solution Steps

Step 1: Calculate the Mean

The mean (expected value) of the probability distribution is calculated using the formula:

\[ \mu = \sum (x \cdot P(x)) = 3 \times 0.37 + 6 \times 0.1 + 7 \times 0.08 + 9 \times 0.13 + 10 \times 0.07 + 15 \times 0.11 + 20 \times 0.14 \]

Calculating this gives:

\[ \mu = 8.59 \]

Step 2: Calculate the Variance

The variance \( \sigma^2 \) is calculated using the formula:

\[ \sigma^2 = \sum ((x - \mu)^2 \cdot P(x)) = (3 - 8.59)^2 \times 0.37 + (6 - 8.59)^2 \times 0.1 + (7 - 8.59)^2 \times 0.08 + (9 - 8.59)^2 \times 0.13 + (10 - 8.59)^2 \times 0.07 + (15 - 8.59)^2 \times 0.11 + (20 - 8.59)^2 \times 0.14 \]

Calculating this gives:

\[ \sigma^2 = 35.34 \]

Step 3: Calculate the Standard Deviation

The standard deviation \( \sigma \) is the square root of the variance:

\[ \sigma = \sqrt{\sigma^2} = \sqrt{35.34} \approx 5.94 \]

Final Answer

The results are as follows:

  • Mean (Expected Value): \( \mu = 8.59 \)
  • Standard Deviation: \( \sigma = 5.94 \)

Thus, the final boxed answers are:

\[ \boxed{\mu = 8.59} \] \[ \boxed{\sigma = 5.94} \]

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