## Question

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....