Explanation Given: The probability density function is, f ( x ) = { 24 x 4 if x ≥ 2 0 otherwise . Formula used: Let f ( x ) be the probability density function for a continuous random variable X . Expected value or mean ( µ ): E ( x ) = ∫ − ∞ ∞ x f ( x ) d x . Variance: V ( x ) = ∫ − ∞ ∞ ( x − μ ) 2 f ( x ) d x . Standard deviation ( σ ): σ = v ( x ) . If f is continuous over the indicated interval and the limit exists, then ∫ a ∞ f ( x ) d x = lim b → ∞ ∫ a b f ( x ) d x . Calculation: By using the above formula, obtain the mean as follows. E ( x ) = ∫ − ∞ ∞ x f ( x ) d x = ∫ 2 ∞ x 24 x 4 d x = lim R → ∞ ∫ 2 R 24 x 3 d x = lim R → ∞ [ − 24 2 ( 1 x 2 ) | 2 R ] On further simplification, E ( x ) = lim R → ∞ [ − 24 2 ( 1 R 2 − 1 ( 2 ) 2 ) ] = lim R → ∞ [ − 24 2 ( 1 R 2 − 1 4 ) ] = [ − 24 2 ( − 1 4 ) ] = 3 Thus, the expected value is μ = 3

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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 expected value, variance and standard deviation for the given probability density function.

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