 # Sums of Random Variables

## Question

First random variable, let's say X, has normal distribution with 0(zero) mean and unit variance.
Second random variable, let's say Y, has lognormal distribution with 0(zero) mean and unit variance.
W=X+Y, what is the distribution of W? Write a detailed explanation.

## Solution Preview

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First a secret: While the sum of random variables is well defined (usually), to actually determine its distribution by a simple formula is usually impossible.
Your example where X is normal and Y is lognormal is such an example, as well shall see. However, in many instances it not necessary to compute the distribution function. For example, if U=X+Y, then E(U)=E(X)+E(Y). So at least expectation and often more moments of U can be computed directly from knowledge of X and Y.

Let X and Y be random variables. Assume for now that X and Y both have finitely many values, say X=1,.., n, and Y=1,..,m and so that the joint probability P(X = i, Y = j) is defined. Recall that the marginal P(X = i) is defined as Σⱼ P(X = i, Y = j)

Let U be the random variable X + Y . U has possible outcomes: i+j, where i runs from 1 to n and j runs from 1 to m. U therefore has values between 2 (the lowest value in this case) and m+n (the highest value in this case). Now many outcomes occur multiple times, whenever i₁ + j₁ = i₂ + j₂....

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