# Lecture - Probability MT22, XIII

### Flashcards

How could you write the definition of the CDF $F _ X$ in the discrete case?

What is the probability density function of $X$ if it has distribution $X \sim \text{Unif}([a,b])$?

What is the probability density function of $X$ if it has distribution $X \sim \text{Exp}(\lambda)$? (careful!)

What is the probability density function of $X$ if it has distribution $X \sim N(\mu, \sigma^2)$?

How is the standard normal distribution $Z$ distributed?::

\[Z \sim N(0, 1)\]What is the notation for the CDF of the standard normal distribution, $F _ Z(z)$?

What, in terms of an integral, is the formula for the CDF of the standard normal distribution $\Phi(z)$?

Given a continuous random variable $X$ with PDF $f _ X(x)$, and $Y = aX + b$, what’s the first step in determining the PDF $f _ Y(y)$?

Rewriting the CDF in terms of $X$:

\[F_Y(y) = F_X\left(\frac{y-b}{a}\right)\]You have a continuous random variable $X$ with PDF $f _ X(x)$, and $Y = aX + b$. You’ve written the CDF in terms of $X$ like $F _ Y(y) = F _ X\left(\frac{y - b}{a}\right)$. How do you now determine $f _ Y(y)$?

Differentiate $F _ X\left(\frac{y-b}{a}\right)$ with respect to $y$.