Question

Suppose that X₁and X₂ are independent and identically distributed normal random variableN(0, 2)

(a) Let Y₁ = X₁ + X₂ and Y₂ = X₁ - X₂ Find the joint probability density function of Y₁ and Y₂

(b) Let Y₁=X ₁ + Box X₂ and Y₂ = X₁ + X₂ Find the joint probability density function ofand Y₂

Answer 1

To find the joint probability density function of Y₁ = X₁ + X₂ and Y₂ = X₁ - X₂, we can start by finding the individual probability density functions of Y₁ and Y₂, and then calculate their joint probability density function using the convolution formula for independent random variables.

Step-by-step explanation:

In this question, we have two independent and identically distributed normal random variables, X₁ and X₂, with a mean of 0 and a standard deviation of 2. We are asked to find the joint probability density function of Y₁ = X₁ + X₂ and Y₂ = X₁ - X₂.

(a) To find the joint probability density function of Y₁ and Y₂, we can start by finding the individual probability density functions of Y₁ and Y₂, and then calculate their joint probability density function. The individual probability density functions can be found using the convolution formula for independent random variables.

(b) To find the joint probability density function of Y₁ and Y₂ when Y₁ = X₁ + Box X₂ and Y₂ = X₁ + X₂.