Question

Suppose the cumulative distribution of the random variable x is f(x) = 0 for x < 0, 0.2x for 0 ≤ x < 5, and 1 for x ≥ 5. What is the cumulative distribution function (CDF) of x?

Answer 1

The cumulative distribution function (CDF) of the given random variable x can be calculated by integrating the probability density function (PDF) over the appropriate ranges. For the given function f(x), the CDF would be 0.1x² for 0 ≤ x < 5, and x - 5 for x ≥ 5.

Step-by-step explanation:

The cumulative distribution function (CDF) of the random variable x can be calculated by integrating the probability density function (PDF) f(x) over the range from negative infinity to x. For the given function f(x):

f(x) = 0 for x < 0

f(x) = 0.2x for 0 ≤ x < 5

f(x) = 1 for x ≥ 5

To find the CDF, we can integrate the PDF:

CDF(x) = ∫ f(t) dt

For 0 ≤ x < 5, the integration yields:

CDF(x) = ∫ 0.2t dt = 0.1t^2 | from 0 to x

For x ≥ 5, the integration yields:

CDF(x) = ∫ 1 dt = t | from 5 to x

The final CDF(x) will be:

CDF(x) = 0.1x² | from 0 to x for 0 ≤ x < 5

CDF(x) = x - 5 | from 5 to x for x ≥ 5

This is the cumulative distribution function of x.