Which of the following conditions must be satisfied for a function $F_{X,Y}(x,y)$ to be a valid joint cumulative distribution function (CDF) for continuous random variables $X$ and $Y$?
Question 2
Given two continuous random variables $X$ and $Y$ with a joint probability density function $f_{X,Y}(x,y)$, which of the following expressions correctly represents the conditional expectation of $Y$ given $X=x$, denoted as $E[Y|X=x]$?
Question 3
In the context of multivariate transforms, if $X_1, X_2, \dots, X_n$ are independent and identically distributed (i.i.d.) random variables, and we define a new variable $S_n = \sum_{i=1}^{n} X_i$, which theorem is crucial for approximating the distribution of $S_n$ as $n \to \infty$?
Question 4
Consider a bivariate normal distribution for random variables $X$ and $Y$. Which of the following statements is true regarding their independence?
Question 5
If $X$ and $Y$ are two random variables with joint probability density function $f_{X,Y}(x,y)$, and we are interested in the distribution of $Z = Y/X$, which of the following methods is most appropriate for finding the PDF of $Z$?
