(a) To determine To show: The function f ( x ) = e − x is a probability density function on the interval [ 0 , ∞ ) and find the probability P ( 0 ≤ X ≤ 1 ) . 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 ) = e − x 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 = ∫ 0 ∞ e − x d x = lim a → ∞ ∫ 0 a e − x d x = lim a → ∞ [ − e − x ] 0 a ( ∵ ∫ e m x d x = e m x m + C , m ∈ N ) = lim a → ∞ (b) To determine To find: The probability P ( 1 ≤ X ≤ 2 ) . (c) To determine To find: The value of the probability P ( X ≤ 2 ) .

Calculus with Applications Books a la Carte Edition

11th Edition

ISBN: 9780133864564

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

Publisher: Pearson Education

See similar textbooks

Concept explainers

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could co…

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a…

Videos

Question

Chapter 11.1, Problem 30E

(a)

To determine

To show: The function f(x)=e−x 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≤2) .

(c)

To determine

To find: The value of the probability P(X≤2) .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11.1, Problem 29E

Chapter 11.1, Problem 31E

Students have asked these similar questions

Find the total area of the shaded regions. y 18- 16- 14- 12- 10- 8- 6- y=ex+1-e 4- 2- 0- 2 3 4 5 -2 -4- X ☑ The total area of the shaded regions is (Type an integer or decimal rounded to three decimal places as needed.)

The graph of f(x), shown here, consists of two straight line segments and two quarter circles. Find the 19 value of f(x)dx. 小 Srxdx. 19 f(x)dx y 7 -7 2 12 19 X ☑

Can you solve this two numerical method eqn and teach me.

Chapter 11 Solutions

Calculus with Applications Books a la Carte 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...

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.