Question

Let X be a random variable whose range is [0, 1] and whose range is given by δ(x) = Cx4. (a) Given the fact that ∫ 1 0 δ(x) dx = 1. Find the constant C. (b) Compute E[X].

Answer 1

To find the constant C, evaluate the integral of δ(x) over the range [0, 1] and set it equal to 1. Then, find the expected value (E[X]) by substituting the expression for δ(x) into the integral and integrating with respect to x.

Step-by-step explanation:

a. To find the constant C, we need to evaluate the integral of δ(x) over the range [0, 1] and set it equal to 1. So, we have ∫10 Cx4 dx = 1. Integrate the function to get C/5 = 1. Solve for C to find C = 5.

b. The expected value (E[X]) of a continuous random variable is given by ∫10 xδ(x) dx. Substitute the expression for δ(x) into the integral and integrate with respect to x. So, we have E[X] = ∫10 5x5 dx. Integrate to find E[X] = 1.