Explanation Given: The probability density function is, f ( x ) = { 1 4 if 0 ≤ x ≤ 1 1 2 if 1 < x ≤ 2 1 4 if 2 < x ≤ 3 0 if otherwise } Formula used: Cumulative Distribution function : “If f is a probability density function, then the cumulative distribution F ( x ) = ∫ − ∞ x f ( t ) d t ”. Calculation: In order to obtain the cumulative distribution function by computing the definite integral with lower limit as −∞ and upper limit x as follows. From the given, probability density function f ( x ) = 0 for x < 0 . Compute the cumulative distribution function for x < 0 . F ( x ) = ∫ − ∞ x f ( t ) d t = ∫ − ∞ x ( 0 ) d t = 0 That is, F ( x ) = 0 for x < 0 . From the given, probability density function f ( x ) = 0 for x < 0 and f ( x ) = 1 4 for 0 ≤ x ≤ 1 . Compute the cumulative distribution function for 0 ≤ x ≤ 1 . F ( x ) = ∫ − ∞ x f ( t ) d t = ∫ − ∞ 0 ( 0 ) d t + ∫ 0 x 1 4 d t = 0 + [ t 4 ] 0 x = x 4 That is, F ( x ) = x 4 for 0 ≤ x ≤ 1 . From the given, probability density function f ( x ) = 0 for x < 0 , f ( x ) = 1 4 for 0 ≤ x ≤ 1 and f ( x ) = 1 2 for 1 < x ≤ 2 . Compute the cumulative distribution function for 1 < x ≤ 2 . F ( x ) = ∫ − ∞ x f ( t ) d t = ∫ − ∞ 0 ( 0 ) d t + ∫ 0 1 1 4 d t + ∫ 1 x 1 2 d t = 0 + [ t 4 ] 0 1 + [ t 2 ] 1 x = 1 4 + x 2 − 1 2 That is, F ( x ) = x 2 − 1 4 for 1 < x ≤ 2 . From the given, probability density function f ( x ) = 0 for x < 0 , f ( x ) = 1 4 for 0 ≤ x ≤ 1 , f ( x ) = 1 2 for 1 < x ≤ 2 and f ( x ) = 1 4 for 2 < x ≤ 3

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Chapter 11.2, Problem 54E

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To find: The cumulative distribution function.

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Chapter 11.2, Problem 53E

Chapter 11.2, Problem 55E

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Chapter 11 Solutions

EP CALCULUS F/BUS.,ECON.-BRIEF-ACCESS

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.

Solution Summary: The author calculates the cumulative distribution function by computing the definite integral with lower limit as and upper limit x.

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