(a) 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 Calculation: Consider, the given function f ( x ) = ( 1 2 ) e − x 2 in the interval [ 0 , ∞ ) . It is observed that, condition 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 ∞ e − x 2 d x = lim a → ∞ 1 2 ∫ 0 a e − x 2 d x = 1 2 lim a → ∞ [ e − x 2 ( − 1 2 ) ] 0 a ( ∵ ∫ e m x d x = e m x m + C , m ∈ N ) = 1 2 ( − 2 ) lim a → ∞ [ e − x 2 ] 0 a On further simplification the above expression becomes, ∫ 0 ∞ f ( (b) To determine To find: The probability P ( 1 ≤ X ≤ 3 ) . (c) To determine To find: The probability P ( X ≥ 2 ) .

Calculus With Applications, Books a la Carte Plus MyLab Math Package (11th Edition)

11th Edition

ISBN: 9780133886849

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 11.1, Problem 31E

(a)

To determine

To show: The function f(x)=(12)e−x2 is a probability density function on the interval [0,∞) and find the probability P(0≤X≤1) .

(b)

To determine

To find: The probability P(1≤X≤3) .

(c)

To determine

To find: The probability P(X≥2) .

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Chapter 11.1, Problem 30E

Chapter 11.1, Problem 32E

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Chapter 11 Solutions

Calculus With Applications, Books a la Carte Plus MyLab Math Package (11th Edition)

Ch. 11.1 - YOUR TURN 1 Repeat Example 1(a) for the function...Ch. 11.1 - Prob. 2YT Ch. 11.1 - Prob. 3YT Ch. 11.1 - Prob. 4YT Ch. 11.1 - Prob. 1WE Ch. 11.1 - Prob. 2WE Ch. 11.1 - Prob. 3WE Ch. 11.1 - Decide whether the functions defined as follows...Ch. 11.1 - Decide whether the functions defined as follows...Ch. 11.1 - Decide whether the functions defined as follows...

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The function f ( x ) = ( 1 2 ) e − x 2 is a probability density function on the interval [ 0 , ∞ ) and find the probability P ( 0 ≤ X ≤ 1 ) .

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To show: The function f(x)=(12)e−x2 is a probability density function on the interval [0,∞) and find the probability P(0≤X≤1) .

To find: The probability P(1≤X≤3) .

To find: The probability P(X≥2) .

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Chapter 11 Solutions