Probability Expected Value

Question

Can someone pls explain the following question in detail with all the steps?

If X is a continuous random variable with density function f (x) that is an even function and E(X) exists, show that E(X) = 0

Answer 1

The expected value is the sum of the x-values times the f(x) values.

An "even function" is defined as f(x) = f(-x). Here are some POSSIBLE and arbitrary values of x and f(x)

x f(x)

1 5

-1 5

9 8

-9 8

27 6

-27 6

Notice that f(x) is the same whether x is a positive or negative value. That's the result of f(x) being an even function.

So, IF these were values of x and f(x), let's calculate the expected value:

EV = (1)(5) + (-1)(5) + (9)(8) + (-9)(8) + (27)(6) + (-27)(6) = 0

Although my choices for the values of x and f(x) are completely arbitrary, it is given that X is a continuous random variable. That means there will be as many values of x as there are -x and since EV = ∑x•f(x) we can expect that every x•f(x) will be "zeroed out" by an opposite (-x)•f(x).

I hope this helps.