Random variable

In probability theory, a random variable (RV) assigns a numerical value to each outcome in a sample space. A random variable is discrete if its possible values form a discrete set. Continuous random variables contains an interval where all points exist, i.e., its probabilities are given by areas under a curve.

We denote the sample space $S$ as the domain of the random variable, and the set $S_X$ as the range of the random variable. Then, the way the random variable is distributed is called a probability distribution.

Functions

For a random variable $X$, and a real-valued function $g(x)$, we define $Y=g(X)$ as the function evaluated at the random variable’s value. $Y$ too is a random variable.

The probability of an event $C$ (involving $Y$) is equal to the probability of the equivalent event $B$ of values of $X$ such that $g(X)$ is in $C$:

$$P(Y\text{ in }C)=P(g(X)\text{ in }C)=P(X\text{ in }B)$$

We can infer from this that:

• The event $g(X)\le y$ is used to find the CDF of $Y$ directly.
• The event $y\le g(X)\le y+h$ is used to find the PDF of $Y$.

We obtain the PDF of $Y$ by differentiating with respect to the input of $F$ (usually $y$). Our differentiation usually has an $F_X$ term within it.

Multiple random variables

The joint PMF for discrete RVs is given by:

$$P_{X,Y}(x,y)=P(X=x\wedge Y=y)$$

And as usual, the sum of all the joint PMFs is 1:

$$\sum _{\forall x}\sum _{\forall y}{P}_{X,Y}(x,y)=1$$

The marginal PMF of $X$ is defined as if we only take one of these sums:

$$\sum _{\forall y}{P}_{X,Y}(x,y)=\sum _{i=0}^{n}P(X=x\wedge Y=y_i)=P(X=x)=P_X(x)$$

The joint CDF for continuous RVs is given by:

$$F_{X,Y}(x,y)=P(X\le x\wedge Y\le y)={\int }_{-\infty }^x{\int }_{-\infty }^y{f}_{X,Y}(x,y)dxdy$$ $$F_{X,Y}(\infty ,\infty )=P(X\le \infty \wedge Y\le \infty )=1$$

The marginal CDF is similarly defined as the marginal PMF:

$$F_{X,Y}(x,\infty )=P(X\le x)=F_X(x)$$

The PDF can be obtained from the CDF by differentiation:

$$f_{X,Y}(x,y)=\frac{{\partial }^2{F}_{X,Y}(x,y)}{\partial x\partial y}$$

Independence

$X$ and $Y$ are independent random variables if any event $A\in X$ is independent of any event $B\in Y$, i.e.,:

$$P(X\text{ in }A,Y\text{ in }B)=P(X\text{ in }A)P(Y\text{ in }B)$$

If $X$ and $Y$ are independent, this implies that:

$$p_{X,Y}(x_j,y_k)=p_X(x_j)p_Y(y_k)$$

i.e., the joint pmf is equal to the product of the marginal pmfs.

The same holds true for the continuous case. For independent random variables $X,Y$, their joint cdf is equal to the product of its marginal cdfs.

$$F_{X,Y}(x,y)=F_X(x)F_Y(y)$$

They’re also independent if and only if their joint pdf is equal to the product of the marginal pdfs:

$$f_{X,Y}(x,y)=f_X(x)f_Y(y)$$