Explanation Given: The function f ( x ) = { 6 x − 6 x 2 if 0 ≤ x ≤ 1 0 otherwise . Definition used: If f is a probability density function, then the associated cumulative distribution function F is defined by F ( x ) = P ( X ≤ x ) = ∫ − ∞ x f ( t ) d t . Furthermore P ( c ≤ X ≤ d ) = F ( d ) − F ( c ) . Calculation: The cumulative distribution function of the probability density function f ( x ) = { 6 x − 6 x 2 if 0 ≤ x ≤ 1 0 otherwise of the random variable X is calculated as follows. First consider if x < 0 . Use the definition of the cumulative distribution function and obtain the cumulative distribution function as follows. F ( x ) = P ( X < 0 ) = ∫ − ∞ x 0 d t [ ∵ f ( x ) = { 6 x − 6 x 2 if 0 ≤ x ≤ 1 0 otherwise ] = 0 Hence, the cumulative distribution function if x < 0 is F ( x ) = 0 . Similarly, if 0 ≤ x ≤ 1 , then F ( x ) = P ( X ≤ x ) = ∫ − ∞ x f ( t ) d t = ∫ − ∞ 0 0 d t + ∫ 0 x f ( t ) d t [ ∵ f ( x ) = { 6 x − 6 x 2 if 0 ≤ x ≤ 1 0 otherwise ] = 0 + ∫ 0 x ( 6 t − 6 t 2 ) d t Simplified further as, F ( x ) = 0 + ∫ 0 x ( 6 t − 6 t 2 ) d t = 6 [ t 2 2 − t 3 3 ] 0 x = 6 ( x 2 2 − x 3 3 ) − 6 ( 0 − 0 ) = 3 x 2 − 2 x 3 Thus, the cumulative distribution function if 0 ≤ x ≤ 1 is F ( x ) = 3 x 2 − 2 x 3

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Chapter 11.2, Problem 4MP

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To find: The cumulative distribution function for the probability density function in Matched problem 1, and use it to compute P(0.3≤X≤0.7).

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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 cumulative distribution function for the probability density function in Matched problem 1, and use it to compute P ( 0.3 ≤ X ≤ 0.7 ) .

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To find: The cumulative distribution function for the probability density function in Matched problem 1, and use it to compute P(0.3≤X≤0.7).

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