Probability MT22, Expectations

@Define the expectation $\mathbb{E}[X]$ of a discrete random variable $X$.

How can you think about the formula

?

What’s a trick for finding $\mathbb{E}[X^2]$ if you know $\mathbb{E}[X]$ and $\mathrm{var}(X)$?

@State the formula for $\mathbb{E}[Y]$ where $Y := h(X)$, $h : \mathbb{R} \to \mathbb{R}$, and $X$ is a discrete random variable.

Suppose $h : \mathbb R \to \mathbb R$ and $X$ is a discrete random variable. @Prove that (∆expectation-of-function-of-rv)

@Define the conditional expectation $\mathbb{E}[X \vert B]$ of a discrete random variable $X$ given an event $B$.

@State the formula for $\mathbb{E}[h(X, Y)]$ for discrete random variables $X, Y$ with joint p.m.f. $p _ {X,Y}$.

How can you rewrite $\mathbb{E}[aX + bY]$?

How can you prove the linearity of expectation $\mathbb{E}[\alpha X + \beta Y] = \alpha\mathbb{E}[X] + \beta\mathbb{E}[Y]$?

@Prove that if $X$ is a discrete random variable and $\mathbb E[X]$ exists, then

• If $X$ non-negative, then $\mathbb E[X] \ge 0$
• If $a, b \in \mathbb R$ then $\mathbb E[aX + b] = a\mathbb E[X] + b$

@Prove the partition theorem for expectations (∆partition-theorem-for-expectations): suppose $B _ 1, B _ 2, \ldots$ is a partition of $\Omega$ such that $\mathbb P(B _ i) > 0$ for all $i \ge 1$. Then for a discrete random variable $X$,

Suppose $X$ and $Y$ are discrete random variables and $a, b \in \mathbb R$ are constants. @Prove that (∆linearity-of-expectation)

@Prove that if $X$ and $Y$ are independent discrete random variables whose expectations exist, then (∆independent-product-expectation)

@Define the expectation $\mathbb{E}[X]$ of a continuous random variable $X$ with density function $f _ X$.

@State the formula for $\mathbb{E}[h(X)]$ for a continuous random variable $X$ with PDF $f _ X$ and $h:\mathbb{R} \to \mathbb{R}$ (provided the absolute version converges).

For a continuous random variable $X$ with PDF $f _ X$, and $h:\mathbb{R} \to \mathbb{R}$, what condition is there for $\mathbb{E}[h(X)]$ to exist?

If $X \ge 0$ and $X$ is a continuous random variable, then what is an equivalent definition of the expectation $\mathbb{E}[X]$ using $\mathbb P(X > x)$?

@Prove that for a non-negative continuous random variable $Z$ (∆continuous-tail-sum-expectation),

Suppose that $X$ is a continuous random variable with p.d.f. $f _ X$. @Prove that if $a, b \in \mathbb R$, $\mathbb E[aX + b] = a\mathbb E[X] + b$ and $\mathrm{var}(aX + b) = a^2 \mathrm{var}(X)$.