Question

"Show that if the probability density function f(x) is symmetric around a number a and the expected value E[X] exists, then E[X] = a."

Answer 1

If the probability density function is symmetric around a number and the expected value exists, then the expected value is equal to that number.

Step-by-step explanation:

To show that if the probability density function f(x) is symmetric around a number a and the expected value E[X] exists, then E[X] = a, we can use the property of symmetry for the expected value. If f(x) is symmetric around a, then the probability density function is the same on either side of a. As a result, the expected value, which is the average value of x weighted by the probability density function, will be equal to a.