(a) To determine To show: The given function is a probability density function and find the probability P ( 0 ≤ X ≤ 1 ) . Explanation Given: The function is, f ( x ) = { 20 x 4 9 if 0 ≤ x ≤ 1 20 9 x 5 if x > 1 , on the interval [ 0 , ∞ ) . Definition used: (a). The function f is a probability density function of a random variable X in the interval [ a , b ] if 1. f ( x ) ≥ 0 for all x in the interval [ a , b ] . 2. ∫ a b f ( x ) d x = 1 (b). Probability density function on ( − ∞ , ∞ ) . If f is a probability density function for a random variable X in the interval ( − ∞ , ∞ ) , then 1. P ( X ≤ b ) = P ( X < b ) = ∫ − ∞ b f ( x ) d x 2. P ( X ≥ a ) = P ( X > a ) = ∫ a ∞ f ( x ) d x 3. P ( − ∞ < X < ∞ ) = ∫ − ∞ ∞ f ( x ) d x = 1 Calculation: Consider, the given function f ( x ) = { 20 x 4 9 if 0 ≤ x ≤ 1 20 9 x 5 if x > 1 , on the interval [ 0 , ∞ ) . It is observed that, condition 1 holds from the definition (a). That is, f ( x ) ≥ 0 for the interval [ 0 , ∞ ) . Next show that condition 2 holds. ∫ 0 ∞ f ( x ) d x = ∫ 0 1 20 x 4 9 d x + ∫ 1 ∞ 20 9 x 5 d x = 20 9 ∫ 0 1 x 4 d x + lim a → ∞ 20 9 ∫ 1 a 1 x 5 d x = 20 9 [ x 5 5 ] 0 1 + 20 9 lim a → ∞ [ − 1 4 x 4 ] (b) To determine To find: The probability P ( X ≥ 1 ) . (c) To determine To find: The probability P ( 0 ≤ X ≤ 2 ) .

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Chapter 11.1, Problem 34E

(a)

To determine

To show: The given function is a probability density function and find the probability P(0≤X≤1) .

(b)

To determine

To find: The probability P(X≥1) .

(c)

To determine

To find: The probability P(0≤X≤2) .

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Chapter 11.1, Problem 33E

Chapter 11.1, Problem 35E

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