Please answer both parts of the question thanks ### Educational Content: Understanding Joint Probability and Gamma Distribution#### Problem 1**Independent Random Variables and Joint Probability**Consider two independent random variables \( X_1 \) and \( X_2 \), each having a standard normal distribution. Define the following:- \( Y_1 = X_1 \)- \( Y_2 = X_1 + X_2 \)**Tasks:**1. Find the joint probability density function (p.d.f.) of \( Y_1 \) and \( Y_2 \).2. Calculate the mean and the variance of \( Y_2 \).#### Problem 2**Gamma Distribution and Moment-Generating Functions**Suppose you have \( n \) independent random variables, each following a gamma distribution with parameters \( \alpha \) and \( \beta \).**Tasks:**1. Determine the moment-generating function of their sum \( Y \).2. Find the probability density function (p.d.f.) of \( Y \).

and are independent random variables having the standard normal distribution.The random variables and are defined as,We can express in terms of ,The PDF of standard normal variable is given as,The PDF of is given by,Calculation for the Jacobian of transformation,The joint PDF of is a product of the marginal PDF of (since, they are independent variables).The joint PDF of is given as,Standard normal variate has a mean and variance of 1.The mean of is,The variance of is, Note:for independent random variables,The joint pdf of ,The mean and variance of are and are respectively.