(a) To determine To show: The function f ( x ) = 1 2 ( 1 + x ) − 3 2 is a probability density function on the interval [ 0 , ∞ ) and find the probability P ( 0 ≤ X ≤ 2 ) . Explanation 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 Proof: Consider, the given function f ( x ) = 1 2 ( 1 + x ) − 3 2 in 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 = 1 2 ∫ 0 ∞ ( 1 + x ) − 3 2 d x = lim a → ∞ 1 2 ∫ 0 a ( 1 + x ) − 3 2 d x = 1 2 lim a → ∞ [ ( 1 + x ) − 3 2 + 1 − 3 2 + 1 ] 0 a ( ∵ ∫ x n d x = x n + 1 n + 1 + C ) = 1 2 lim a → ∞ [ ( 1 + x ) − 1 2 ( − 1 2 ) ] 0 a On further simplification the above expression becomes, ∫ 0 ∞ f ( x ) d x = 1 (b) To determine To find: The probability P ( 1 ≤ X ≤ 3 ) . (c) To determine To find: The probability P ( X ≥ 5 ) .

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Chapter 11.1, Problem 29E

(a)

To determine

To show: The function f(x)=12(1+x)−32 is a probability density function on the interval [0,∞) and find the probability P(0≤X≤2) .

(b)

To determine

To find: The probability P(1≤X≤3) .

(c)

To determine

To find: The probability P(X≥5) .

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Chapter 11.1, Problem 28E

Chapter 11.1, Problem 30E

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