(a) To determine To show: The given function is a probability density function and find the probability P ( 0 ≤ X ≤ 2 ) . Explanation Given: The function is, f ( x ) = { x 3 12 if 0 ≤ x ≤ 2 16 3 x 3 if x > 2 . The interval is [ 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 ) = { x 3 12 if 0 ≤ x ≤ 2 16 3 x 3 if x > 2 , 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 2 x 3 12 d x + ∫ 2 ∞ 16 3 x 3 d x = 1 12 ∫ 0 2 x 3 d x + lim a → ∞ 16 3 ∫ 2 a 1 x 3 d x = 1 12 [ x 4 4 ] 0 2 + 16 3 lim a → ∞ [ − 1 2 x 2 ] 2 a (b) To determine To find: The probability P ( X ≥ 2 ) . (c) To determine To find: The probability P ( 1 ≤ X ≤ 3 ) .

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Question

Chapter 11.1, Problem 33E

(a)

To determine

To show: The given function is a probability density function and find the probability P(0≤X≤2) .

(b)

To determine

To find: The probability P(X≥2) .

(c)

To determine

To find: The probability P(1≤X≤3) .

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Chapter 11.1, Problem 32E

Chapter 11.1, Problem 34E

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