02 Inferences about distributions of function(RV)

Y = g(X).

Use $P r (g (X) \in A) = P r (X \in g^{- 1} (A))$ . So, given CDF, PDF of X, can deduce CDF of g(X) and thence derive PDF of g(X).

Using $\frac{dg^{-1}(Y)$

If g is monotone in $(x, x + δ x)$ : $p_{X} (x) δ x \approx p_{Y} (y) δ y$ , taking $(x, x + δ x)$ to $(y, y + δ y)$ using g: So $p_{Y} (y) = p_{X} (x) | \frac{d x}{d y} | = p_{X} (g^{- 1} (y)) | \frac{d g^{- 1} (y)}{d y} |$ : so maximum probability density changes with variable change.

If g is not continuous, but $\exists$ partition $A_{0}, . . A_{k}$ with $P r (X \in A_{0}) = 0$ , with ${g_{i}} = g$ over ${A_{i}}$ monotone; then $p_{Y} (y) = \sum_{i} p_{X} (g^{- 1} (y)) | \frac{d g_{i}^{- 1} (y)}{d y} |$ ; where $\sum$ appears to account for the probability that Y=y over various domains of X.

Extension to multidimensional distributions

$Y = g (X_{1}, X_{2}); X_{1} = h (Y, X_{2}) $. F i x $ X_{2} = x_{2} $; g e t $ p (Y, x_{2}) = p_{X_{1}, X_{2}} (X_{1} = h^{- 1} (Y, x_{2}) | x_{2}) | \frac{d h^{- 1} (Y, x_{2})}{d Y} | $; t h e n d o $ p_{Y} (y) = \int p (Y, x_{2}) d x_{2}$ .

Using moment generating functions

Given $m_{X} (t)$ , find \ $m_{Y} (t) = E [e^{f (X) t}]$ ; thence deduce pdf of Y.

Using \(\frac{dg^{-1}(Y)\)

Extension to multidimensional distributions

Using moment generating functions