# What is the distribution of conditional expectation of a function f(X) of the random variable X? i.e. E(f(X)|X)

I have a continuous random variable X with a known PDF. I want to find the distribution of f(X) where f(X) is a function of X (e.g. X^2+5X). How do I find the distribution of E(f(X)|X))?

I know how to approach this when X is discrete. In this case, I would simply calculate E(f(X)|X=t) by summing over all possible values of X=t and then replace t by X to get the distribution of E(f(X)|X).

• It sounds like you need a formula for transforming a variable. The hits at this search ought to help: stats.stackexchange.com/search?q=jacobian+change+score%3A1. – whuber Mar 21 '19 at 16:20
• What do you mean by E(f(X)|X)). Also, what do you mean by lowercase x as opposed to the uppercase X? – Martin Drozdik Mar 21 '19 at 16:24
• lower case x was a typo and is fixed. By E(f(X)|X), I mean conditional expectation of f(X) given X. I think it should be f(X) but I just wanted to confirm. Then as @whuber mentioned, I will have to look at deriving distribution of function of a random variable. – exAres Mar 21 '19 at 16:27

## 1 Answer

for any random variable(discrete , continues and mixture) and Borel function, f

$$E(f(X)|X)=f(X)$$ so you just need to compute distribution of $$f(X)$$

so $$E(X^2+X|X)=X^2+X$$

note $$E(f(X)|X)=f(X)$$ is a random variable and $$E(f(X)|X=t)=f(t)$$ is a number

this property ,$$E(f(X)|X)=f(X)$$ come from defination of conditional expectation and note that $$\sigma(f(X)) \subset \sigma(X)$$