Aquantity x has cumulative distribution function p(x) = x − x2/4 for 0 ≤ x ≤ 2 and p(x) = 0 for x < 0 and p(x) = 1 for x > 2. find the mean and median of x.

The mean of x is 2/3, and the median of x is

The cumulative distribution function is given as:

Differentiate the cumulative distribution function.

So, we have:

The mean is then calculated as:

So, we have:

Expand

Integrate

Expand

Reduce fraction

So, the mean of x is 2/3

To calculate the median, we have:

Rewrite as:

The median element at the mid-position (i.e. 1/2).

So, we have:

Multiply through by 4

Rewrite as:

Solve for M

Recall that:

The domain is given as:

is outside the domain.

Hence, the median of x is

Read more about mean and median at:

link

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step-by-step explanation: