Random Variable

Random variable is a function which maps the outcome of a simple random experiment to a real number. denoted by capital italized roman characters like $X$ or $Y$.

Suppose that a coin is tossed twice so that the sample space is $S=HH,HT,TH,TT$. Let $X$ represent the number of heads that can come up. With each sample point we can associate a number for $X$ as shown below.

Sample point	HH	HT	TH	TT
X	2	1	1	0

Types of Random Variable

• Discrete
  • can take finite or countably infinite number of values
  • probability distribution PMF : $\{\begin{array}{ll}P(X=x)& \text{if }x\in \Omega \\ 0& \text{if }x\notin \Omega \end{array}$
• Continuous
  • takes on a noncountably infinite number of values
  • probability distribution PDF : $P(a\le X\le b)={\int }_a^{b}f(x)dx$

Probability Function

A probability function is a mathematical function that provides probabilities for the possible outcomes of the random variable, $X$. It is typically denoted as $f(x)$.

1. Probability Mass Function
   A probability mass function (PMF) is a function that gives the probability that a discrete random variable is exactly equal to a certain value. In other words, it maps each possible outcome of a discrete random variable to its probability of occurrence.
   If $X$ is discrete, then $f(x)=P(X=x)$. In other words, the PMF for a constant, $x$, is the probability that the random variable $X$ is equal to $x$.
   Properties of PMF : - $f(x)\ge 0$ for x in sample space otherwise 0. - ${\sum }_x^{}f(x)=1$.

2. Probability Density Function
   If the random variable is a continuous random variable, the probability function is usually called the probability density function (PDF). Contrary to the discrete case,
   $f(x)\ne P(X=x)$ Properties of PDF : - $f(x)\ge 0$ for x in sample space otherwise 0. - Area under the curve is equal to 1.

3. Cumulative Distribution Function
   A cumulative distribution function (CDF), usually denoted $F(x)$, is a function that gives the probability that the random variable, $X$, is less than or equal to the value $x$.

Discrete Probability Distributions

Let $X$ be a discrete random variable, and suppose that the possible values that it can assume are given by $x1,x2,x3,...,$ arranged in some order. Suppose also that these values are assumed with probabilities given by:

$P(X=x_k)=f(x_k),k=1,2,3,4,...$

It is convenient to introduce the probability function, also referred to as probability distribution, given by :

$P(X=x)=f(x)$

For $x=x_k$ this reduces to (1) while for other values of $x,f(x)=0$. In general, $f(x)$ is a probability function if :

1. $f(x)\ge 0$
2. ${\sum }_x^{}f(x)=1$ where the sum is taken over all possible values of $x$.

Expected Value and Variance of a Discrete Random Variable

Expected value or mean of discrete random variable is denoted by $\mu =E(X)=\sum x_{i}f(x_i)$.
Formula basically says multiply each value by its respective probability and add all of them together.

Variance of a discrete random variable is given by $\sigma =Var(X)=(\sum x_i-\mu {)}^{2}f(x_i)$ Formula basically says take each value of x, subtract the expected value, square that value and multiply that value by its probability. Then sum all of those values.

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1. https://www.gs.washington.edu/academics/courses/akey/56008/lecture/lecture4.pdf
2. http://users.stat.umn.edu/~helwig/notes/RandomVariables.pdf
3. http://www.stat.yale.edu/Courses/1997-98/101/ranvar.htm
4. https://www.stat.pitt.edu/stoffer/tsa4/intro_prob.pdf
5. https://byjus.com/maths/random-variable/
6. https://online.stat.psu.edu/stat500/lesson/3/3.1