Question

Suppose the random variable X has the following probability density function: f(x;k,θ)=kθᵏxᵏ⁻¹,x≥θ.

If k>1, compute E(X).

Answer 1

To compute the expected value (E(X)) of a random variable with a given probability density function, we need to multiply each value of X by its probability and add the products. However, in this case, the expected value does not exist for the given probability density function.

Step-by-step explanation:

The random variable X has the probability density function: f(x;k,θ)=kθᵏxᵏ⁻¹,x≥θ, where k > 1. To compute the expected value of X, denoted as E(X), we need to multiply each value of X by its probability and add the products. The formula for E(X) is Σ(x * P(x)). In this case, the values of X are x≥θ, so we need to integrate the probability density function over the range x≥θ. Let's calculate it step by step:

Integrate the probability density function over the range x≥θ: ∫(kθᵏxᵏ⁻¹)dx, from θ to ∞.

Apply the power rule of integration to simplify the integral: kθᵏ∫(xᵏ⁻¹)dx.

Integrate xᵏ⁻¹ with respect to x: kθᵏ * [xᵏ/ᵏ] evaluated from θ to ∞.

Substitute ∞ into the expression: kθᵏ * (∞ᵏ/ᵏ).

Observe that (∞ᵏ) is undefined, so we conclude that the expected value does not exist for this random variable.