# Thread: p.g.f / density function

1. ## p.g.f / density function

(a) A discrete random variable takes on the values 0,1,2 and 3 with probabilities 1/8, 1/4, 1/2 respectively. Determine the p.g.f of the sum of two such random variables which are independent, stating any properties of the p.g.f's that you need to use.

(b) A random variable X has density given by

f(x) = 3x^2 0 < x < 1
0 otherwise

Let Y be a new random variable defined by Y = X^2. Determine the density function of Y.

2. Originally Posted by shtaM
(a) A discrete random variable takes on the values 0,1,2 and 3 with probabilities 1/8, 1/4, 1/2 respectively. Determine the p.g.f of the sum of two such random variables which are independent, stating any properties of the p.g.f's that you need to use.

(b) A random variable X has density given by

f(x) = 3x^2 0 < x < 1
0 otherwise

Let Y be a new random variable defined by Y = X^2. Determine the density function of Y.
(a) Apply the following:

Let $\displaystyle Y = X_1 + X_2$ where $\displaystyle X_1$ and $\displaystyle X_2$ are independent.

Then $\displaystyle G_Y(z) = G_{X_1}(z) \cdot G_{X_2}(z)$ where $\displaystyle G_{X_1}(z) = G_{X_2}(z) = E(z^X) = \sum_{x=0}^3 \Pr(X = 0) \cdot z^x$.

(b) The cdf of $\displaystyle Y$ is given by

$\displaystyle G(y) = \Pr(Y < y) = \Pr(X^2 < y) =\Pr(-\sqrt{y} < X < + \sqrt{y}) = \int_{0}^{\sqrt{y}} 3x^2 \, dx$.

The pdf of $\displaystyle Y$ is given by $\displaystyle g(y) = \frac{dG}{dy}$.