Question

In a continuous distribution, whose probability density function is given by

f(x) = 3/4x(2 – x) , 0 < x < 2. Find the expected value of x.

Answer 1

To find the expected value of x for the continuous distribution with the given probability density function, we can use the formula E(X) = ∫x * f(x) dx. Evaluating the integral and simplifying, we find that the expected value is 1.

Step-by-step explanation:

To find the expected value of x for a continuous distribution with the given probability density function f(x) = (3/4)x(2 - x), we can use the formula: E(X) = ∫x * f(x) dx, where E(X) represents the expected value of x.

First, let's find the integral of the probability density function.

We have: ∫(3/4)x(2 - x) dx = (3/4) * ∫x(2 - x) dx

Using the power rule for integration and distributing the (3/4) term, we get: (3/4) * (∫2x - x^2 dx) = (3/4) * [x^2 - (1/3)x^3] + C

Now, we can evaluate this expression from 0 to 2 to find the expected value:

E(X) = [(3/4) * (2^2 - (1/3)2^3)] - [(3/4) * (0^2 - (1/3)0^3)] = [(3/4) * (4 - 8/3)] - 0

Simplifying further, we have: E(X) = (3/4) * (12/3 - 8/3) = (3/4) * (4/3) = 12/12 = 1

Therefore, the expected value of x for the given continuous distribution is 1.