For any positive number a there is a density on 01 proportio

For any positive number a, there is a density on (0,1) proportional to x^a.

a. Find the density.

b. Find the associated CDF.

Section 3.2, page 167 7. f. For any positive For any positive number a, there is a density on (0, 1) proportional to r a. Find this density. b. Find the associated CDF.

Solution

a)

If

f(x) = k x^a

Then integrating from 0 to 1 for normality,

Integral [k x^a dx]|(0,1)

= k x^(a+1) / (a+1) |(0,1)

= k/(a+1) = 1

Thusm

k = a+1

Thusm

f(x) = (a+1) x^a [ANSWER]

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b)

Thusm the cdf is

Integral (f(x) dx)) = x^(a+1) + C = F(x)

As F(1) = 1 (rightmost endpoint must be 1 for a cdf)

1^(a+1) + C = F(1)

C = 0

Thus,

F(x) = x^(a+1) [ANSWER]