4.4 Random Variables

Random variables

Discrete

Expectation

Variance

Continuous

Mode

Median

Mean

Standard deviation

Probability density function

Probability mass function

Linear Transformations

Interquartile range

Random Variable	Definition
Discrete random variables	A random variable assigns numbers to the possible outcomes of a random experiment. A discrete random variable, $X$ , has a finite or countably infinite set of distinct values. Discrete probability distributions are representable in a table due to its categorical nature. The probabilities must add up to 1. The probability distribution of a discrete random variable describes the probabilities associated with each value. For a random variable $X$ with possible values { $x_1,x_2,...,x_n$ } and corresponding probabilities { $p_1,p_2,...,p_n$ }: • $0\leq p_i\leq 1$ • $\sum_{i=1}^np_i=1$ Example: If $P(X)=\frac x{2+1}$ for $x=1,2,3,4$ , then $P(X)$ is a valid probability mass function.
Expectation (Mean)	The expected value $E(X)$ of a discrete random variable $X$ with values $x_i$ and probabilities $p_i$ : $E(X)=\sum _{i=1}^nx_ip_i$ $E(X)$ is the mean of the probability distribution of $X$ , denoted $\mu$ .
Variance	$Var(X)$ measures the spread of the random variable around its mean: $Var(X)=\sum_{i=1}^np_i(x_i-\mu)^2$ Alternatively: $Var(X)=E(X^2)-(E(X))^2$ Standard deviation $\sigma=\sqrt{Var(X)}$
Linear transformations	• $E(aX+b)= aE(X) + b$ • $E(X + Y) = E(X) + E(Y)$ • $Var(aX+b) = a^2Var(X)$ • $Var(X + Y) = Var(X) + Var(Y)$ if $X$ and $Y$ are independent. • $\sigma(X) = \|a\|\sigma(X)$

Random Variable	Definition
Continuous random variables	A random variable $X$ is continuous if its cumulative distribution function is continuous. If we have: $F_X(x)=\int_{-∞}^xf_X(t) dt$ $f_X$ is the probability density function (PDF) of $X$ . Note that PDF and PMF are fundamentally different objects, as PDFs can be greater than 1. Below, for simplicity, we denote PDF as $f(x)$ . A continuous random variable $X$ can take any value within an interval $[a,b]$ . PDF $f(x)$ describes the probability distribution: • $P(c\leq X\leq d)=\int_c^df(x)dx$ • $f(x)\geq 0$ for all $x$ • $\int_a^bf(x)dx=1$
Mode, median, mean	Mode: The value $x$ at which $f(x)$ is maximum. Median: The value $m$ such that $\int_a^mf(x)dx=\frac 12$ . Mean (Expected Value): $\mu =E(X)=\int_a^bxf(x)dx$
Variance and standard deviation	Variance: $\sigma^2=Var(X)=\int_a^b(x-\mu)^2f(x)dx=\int_a^bx^2f(x)dx-\mu^2$ Standard deviation: $\sigma=\sqrt{Var(X)}$

4.4 Random variables

https://forms.gle/QynSnLyHR8HQ66557

4.4 Random Variables