Lecture - Probability MT22, XIII

Created: November 26, 2022

Flashcards

How could you write the definition of the CDF $F _ X$ in the discrete case?
\[F_X(x) = \sum_{u\in(-\infty, x] \cap \text{Im}X} p_X(u)\]

What is the probability density function of $X$ if it has distribution $X \sim \text{Unif}([a,b])$?
\[f_x(x) = \begin{cases}\frac{1}{b - a} & \text{for } a\le x\le b \\\\0 &\text{otherwise}\end{cases}\]

What is the probability density function of $X$ if it has distribution $X \sim \text{Exp}(\lambda)$? (careful!)
\[f_x(x) = \begin{cases}\lambda e^{-\lambda x}& x \ge 0 \\\\0 & \text{otherwise}\end{cases}\]

What is the probability density function of $X$ if it has distribution $X \sim N(\mu, \sigma^2)$?
\[f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{1}{2\sigma^2}(x-\mu)^2\right)\]

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, $F _ Z(z)$?
\[\Phi(z)\]

What, in terms of an integral, is the formula for the CDF of the standard normal distribution $\Phi(z)$?
\[\Phi(z) = \int^z_{-\infty} \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}\]

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 $y$.

Given a continuous random variable $X$ with density function $f _ X$, what is the mean or expectation $\mathbb{E}[X]$ of $X$?
\[\mathbb{E}[X] = \int^\infty_{-\infty} xf_X(x)\text{d}x\]


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