I'm trying to understand the intuition behind the formula for a random variable which is a function of another random variable. Formula: ${\sum}f(x_i)/g'(x_i)$
I tried to take as example a fair dice: 1,…,6 is coming out at $1/6$. But if we use the formula for $y=x^2$, than we won't get 1/6 for 36 as we expect. Why is that?
Could someone explain me the intuition behind this formula?
Best Answer
Here is a classic example that explains where the formula you mentioned comes from:
Suppose $Y = X^n$ then, by the definition of a CDF: $$F_Y(y) = P(Y \leq y) = P(X^n \leq y) = P(X \leq y^{(1/n)})$$ Therefore, $$F_Y(y) = F_X(y^{(1/n)})$$ and we differentiate (using the chain rule) $$f_Y(y) = \frac {1}{n} y^{\frac {1}{n} -1}*f_X(y^{\frac{1}{n}})$$ Written concisely, this is the formula: $$f_Y(y) = \sum_{n=1}^{i} \frac {f(x_i)}{|g'(x_i)|}$$ Where you solve $Y = g(x)$ for $x_i$