   Chapter 8.5, Problem 5E

Chapter
Section
Textbook Problem

Let f (x) = c/(1 + x2). (a) For what value of c is f a probability density function? (b) For that value of c, find P (−1 < X < 1) .

(a)

To determine

To find: The value of c when the function (f) as a probability density functions.

Explanation

Given information:

The function is f(x)=c1+x2 .

Show the function as follows:

f(x)=c1+x2 (1)

Apply probability density function as follows:

(A) The probability density function f of a random variable X satisfies the condition f(x)0 for all x.

(B) The probabilities are measured on a scale from 0 to 1, it follows that f(x)dx=1 .

f(x)>_0 is satisfied when the value of c0 .

Apply the probability density function (B) in Equation (1) as shown below.

f(x)dx=c1+x2

(b)

To determine

To calculate: The value of P(1X1) .

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