Flashcards
For a continuous random variable $X$ with PDF $f _ X$, and $h:\mathbb{R} \to \mathbb{R}$, what is $\mathbb{E}[h(X)]$ (provided the absolute version converges)?
\[\int^\infty_{-\infty} h(x) f_X(x)\text{d}x\]
For a continuous random variable $X$ with PDF $f _ X$, and $h:\mathbb{R} \to \mathbb{R}$, what condition is there for the $\mathbb{E}[h(x)]$ to exist?
\[\int^\infty_{-\infty} |h(x)| f_X(x)\text{d}x \text{ converges}\]
If $X \ge 0$ and $X$ is a continuous random variable, then what is an equivalent definition of the expectation $\mathbb{E}[X]$ using $\mathbb P(X > x)$?
\[\int^\infty_0 \mathbb P(X > x) \text{d}x\]
For two continuous random variables $X, Y$ then what is $F _ {X,Y}(x,y)$ shorthand for?
\[\mathbb{P}(X \le x, Y \le y)\]
What form does the joint CDF $F _ {X,Y}(x,y)$ have to be for $X$ and $Y$ to be jointly continuously distributed with PDF $f _ {X,Y}$?
\[F_{X,Y}(x,y) = \int^x_{-\infty}\int^y_{-\infty} f_{X,Y}(u,v)\text{d}v\text{d}u\]
For a jointly continuously distributed $(X, Y)$, what is $\mathbb{P}(a < X \le b, c < Y \le d)$ in terms of an integral?
\[\int_{x=a}^b \int_{y=c}^d f_{X,Y}(x,y) \text{d}y\text{d}x\]
For a jointly continuously distributed $(X, Y)$, what is $\mathbb{P}(a < X \le b, c < Y \le d)$ in terms of the joint CDF?
\[F_{X,Y}(b, d) - F_{X,Y}(a,d) - F_{X,Y}(b,c) + F_{X,Y}(a,c)\]
For “nice enough” $A \subseteq \mathbb{R}^2$, what is $\mathbb{P}((X,Y) \in A)$?
\[\iint_{(x,y) \in A} f_{X,Y}(x,y) \text{d}y\text{d}x\]
If $X, Y$ are jointly continuous with joint density function $f _ {X,Y}$, how can you recover $f _ X(x)$?
\[f_X(x) = \int^\infty_{-\infty} f_{X,Y}(x,y) \text{d}y\]
If $X, Y$ are jointly continuous with joint density function $f _ {X,Y}$, how can you recover $f _ Y(x)$?
\[f_Y(x) = \int^\infty_{-\infty} f_{X,Y}(x,y) \text{d}x\]