Chapter 8.5, Problem 3E

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

Chapter
Section

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

# Let f (x) = 30x2(1 − x)2 for 0 ≤ x ≤ 1 and f (x) = 0 for all other values of x. (a) Verify that f is a probability density function. (b) Find P ( X ≤ 1 3 ) .

(a)

To determine

To verify: The function f is a probability density function.

Explanation

Given information:

The function f(x)=30x2(1âˆ’x)2 for 0â‰¤xâ‰¤1 .

The function f(x)=0 for all other values of x.

Hence, f(x)â‰¥0 for all x.

The region lies between a=0 and b=1 .

Show the function as follows:

f(x)=30x2(1âˆ’x)2 (1)

Modify Equation (1).

f(x)=30x2(1+x2âˆ’2x)=30x2+30x4âˆ’60x3=30x4âˆ’60x3+30x2

Apply probability density function as follows:

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

(b)

To determine

To calculate: The value of P(X13) .

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