The mean (one measure of average) of a random variable with pdf on the interval is . Find the mean if .

Question

The mean  (one measure of average) of a random variable with pdf  on the interval  is   . Find the mean if  .

Quizplus · Accepted Answer

The mean of a probability density function on an interval (a,b) is given by:

$\int_a^b x f(x) dx$

In this case, we are given that the mean on the interval (c,d) is e. So we have:

$\int_c^d x f(x) dx = e$

We want to find the mean on the interval (a,b). We can use the same formula with (a,b) instead of (c,d):

$\int_a^b x f(x) dx = ?$

We don't know the value of this integral, but we can use the fact that the integral of the probability density function over any interval is 1:

$\int_a^b f(x) dx = 1$

We can rearrange this to get:

$\int_a^b (x-e) f(x) dx = 0$

Notice that if we integrate $(x-e) f(x)$ over the interval (c,d), we get:

$\int_c^d (x-e) f(x) dx = \int_c^d x f(x) dx - e \int_c^d f(x) dx = e - e = 0$

So we have:

$\int_a^b (x-e) f(x) dx = 0$

This means that the mean on the interval (a,b) is also e. Therefore, the answer is (A).

The Mean (One Measure of Average) of a Random