Explanation Formula used: The mean and standard deviation for the continuous random variable are, (1) Mean μ = ∫ − ∞ ∞ x f ( x ) d x . (2) Standard deviation σ = V ( X ) , where variance V ( X ) = ∫ − ∞ ∞ x 2 f ( x ) d x − μ 2 . Calculation: The probability density function is f ( x ) = { p x p + 1 if x ≥ 1 0 otherwise . By using the above formula, compute the mean with respective probability density function. μ = ∫ − ∞ 1 p x p + 1 x d x + ∫ 1 ∞ p x p + 1 x d x = 0 + p lim b → ∞ ∫ 1 b x − p d x = p lim b → ∞ [ x − p + 1 − p + 1 ] 1 b = p 1 − p lim b → ∞ [ ( b ) − p + 1 − ( 1 ) − p + 1 ] On further simplification. μ = p 1 − p [ − 1 ] = p p − 1 Thus, the value of mean is μ = p p − 1 . In order to find the standard deviation, first compute the variance. V ( X ) = ∫ − ∞ ∞ p x p + 1 x 2 d x − ( p p − 1 ) 2 ( ∵ μ = p p − 1 ) <

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Chapter 11.4, Problem 66E

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To find: The standard deviation of a probability density function and explain the restriction on p in standard deviation.

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Chapter 11.4, Problem 65E

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Ch. 11.1 - Evaluate the following, if it converges: 3dx(x1)2.Ch. 11.1 - Prob. 2MP Ch. 11.1 - Prob. 3MP Ch. 11.1 - Prob. 4MP Ch. 11.1 - Prob. 5MP Ch. 11.1 - Prob. 6MP Ch. 11.1 - Prob. 1ED Ch. 11.1 - Prob. 2ED Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E

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The standard deviation of a probability density function and explain the restriction on p in standard deviation.

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