the histogram shows the distribution of hurricanes that have hit a country from 1851 through 2015, where 1 is the weakest level and 5 is the strongest level.

(a) find the mean, variance, and standard deviation of the probability distribution.
(b) interpret the results.

(a) the mean is 2.068
(type an integer or a decimal. do not round.)

the variance is
(round to two decimal places as needed.)

Question

the histogram shows the distribution of hurricanes that have hit a country from 1851 through 2015, where 1 is the weakest level and 5 is the strongest level.

(a) find the mean, variance, and standard deviation of the probability distribution.
(b) interpret the results.

(a) the mean is 2.068
(type an integer or a decimal. do not round.)

the variance is
(round to two decimal places as needed.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Mean of Probability Distribution", "Variance of Probability Distribution" ], "new_concepts": [], "current_concepts": [ "Mean of Probability Distribution", "Variance of Probability Distribution" ] } ## Define the probability distribution $$ \begin{array}{c|c} x & P(x) \ \hline 1 & 0.407 \ 2 & 0.255 \ 3 & 0.212 \ 4 & 0.115 \ 5 & 0.011 \ \end{array} $$ ## Calculate the mean $\mu$ $$ \begin{aligned} \mu &= \sum x \cdot P(x) \ &= 1(0.407) + 2(0.255) + 3(0.212) + 4(0.115) + 5(0.011) \ &= 0.407 + 0.510 + 0.636 + 0.460 + 0.055 \ &= 2.068 \end{aligned} $$ ## Calculate the variance $\sigma^2$ $$ \begin{aligned} \sum x^2 \cdot P(x) &= 1^2(0.407) + 2^2(0.255) + 3^2(0.212) + 4^2(0.115) + 5^2(0.011) \ &= 1(0.407) + 4(0.255) + 9(0.212) + 16(0.115) + 25(0.011) \ &= 0.407 + 1.020 + 1.908 + 1.840 + 0.275 \ &= 5.450 \ \sigma^2 &= \sum x^2 \cdot P(x) - \mu^2 \ &= 5.450 - (2.068)^2 \ &= 5.450 - 4.276624 \ &= 1.173376 \approx 1.17 \end{aligned} $$ The variance is 1.17 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Variance of Probability Distribution" ] }

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QUESTION IMAGE

Question

Explanation:

Define the probability distribution

Calculate the mean \(\mu\)

Calculate the variance \(\sigma^2\)

Answer: