Suppose that Xi has a normal distribution with mean μi and variance σi^2, i=1,2 . Let X1 and X2

step 1 For exercise 1a, we have m sub y of t equivalent to e to the power of mu sub 1 times t plus sigm

Suppose that Xi has a normal distribution with mean μi and...

Suppose that $X_{i}$ has a normal distribution with mean $\mu_{i}$ and variance $\sigma_{i}^{2}, i=1,2 .$ Let $X_{1}$ and $X_{2}$ be independent.
(a) Find the moment-generating function of $Y=X_{1}+X_{1}$.
(b) What is the distribution of the random variable $Y ?$

Key Concepts

Sum of Normal Random Variables

An important property of the normal distribution is that the sum of independent normal random variables is also normally distributed. The mean of the sum is the sum of the means, and the variance of the sum is the sum of the variances. This result follows both from direct computation using the MGFs and from the convolution of two normal probability density functions.

Moment Generating Function (MGF)

The moment generating function of a random variable provides a convenient way to capture all of its moments. The MGF, defined by M_X(t) = E[e^(tX)], can be used to find the mean, variance, and higher moments by differentiating with respect to t. It uniquely characterizes the probability distribution, provided it exists in an open interval around zero.

MGF of a Normal Distribution

For a normally distributed random variable with mean ? and variance ?², the moment generating function is given by M_X(t) = exp(?t + (1/2)?²t²). This concise form is especially useful because it easily reveals the first two moments (mean and variance) and because the exponential of a quadratic function is stable under multiplication, which is a key property when dealing with sums of normal random variables.

Independence and MGF Multiplication

When two random variables are independent, the MGF of their sum is the product of their individual MGFs. This property, M_{X+Y}(t) = M_X(t) × M_Y(t), simplifies the derivation of the moment generating function of a sum, as it allows one to work with each component separately. This is particularly useful in the context of normal distributions, where the product of the MGFs results in an MGF that corresponds to a normal distribution.

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