# Functions of Random Variables

In what cases would I want to know $\operatorname{E}\left[e^X\right]$? I'm in an introductory probability course and I'm presented with exercises like this often, but I'm wondering when someone would want to know, say the probability density function of a random variable $Y$ for $Y=2X$. When is it important to explore functions of random variables instead of just the random variable itself?

• It's important to evaluate expectations of functions of random variables whenever you're interested in knowing either the expectation of that function of the random variable, or the expectation of the distribution of a random variable transformed in that way (the law of the unconscious statistician makes those two things equivalent). This happens pretty much all the time. For example, people often analyze data on the log-scale (that is transform data by taking logs in order to analyze it). They might then want to discuss what it means on the original scale. ...(ctd) Nov 10 '13 at 0:25
• (ctd)... So if $X$ is the log-scale variable, but they want to talk about the mean of the original variable, they're interested in estimating $\text{E}[e^X]$. Nov 10 '13 at 0:26

1) $MGF_X(t)=E(e^{Xt})$ is the definition of the moment generating function of a random variable, which is a very convenient tool to work with.

2) The statistical estimators, like OLS, MLE, etc, are by construction functions of random variables.

3) Apart from the above, we consider functions of random variables whenever a theoretical model leads us there, or whenever a non-linear transformation induces (or is thought to induce) desirable properties at the technical estimation level. Two examples:

• You want to estimate the parameters of a "Cobb-Douglas" production function, $Q = A K^aL^{1-a}$. ($Q$= quantity, $K$=capital, $L$=labor). You take the logs to transform the problem into a linear one

$$\ln Q = \ln A + a\ln K + (1-a)\ln L + u$$

Now you are dealing with functions of random variables. If you want to consider the estimated error term back in levels you will have to consider $E(e^u)$.

• You want to estimate a binary-choice model, using the logit specification. $Y$ is the 0/1 binary dependent variable, $X$ is a matrix containing the variables that are thought to affect the probability of $Y$ acquiring the value $1$. Your model is

$$P(Y=1\mid X) = \Lambda(X) = \Big( 1+ \exp\{-a-X\beta\}\Big)^{-1}$$

and it contains functions of random variables.