Random Variables on Amit Rajan

Random Variables (Part 3: Jointly Distributed Random Variables)

Tue, 13 Nov 2018 06:11:47 +0100

#### Independent Random Variables :

If $X$ and $Y$ are independent random variables and $S$ and $T$ are sets of numbers then,

\[P(X \in S \ and \ Y \in T) = P(X \in S) P(Y \in T)\]

Hence, for independent random variables $X_1, X_2, ..., X_n$ and constants $c_1, c_2, ..., c_n$, the variance of linear combination $c_1X_1 + c_2X_2 + ... + c_nX_n$ is given as:

\[\sigma _{c_1X_1 + c_2X_2 + ... + c_nX_n}^2 = c_1^2\sigma _{X_1}^2 + c_2^2\sigma _{X_2}^2 + ... + c_n^2\sigma _{X_n}^2\]
#### Independence and Simple Random Samples :

When a simple random sample of numerical values is drawn from a population, each item in the sample can be thought of as a random variable. The items in a simple random sample may be treated as independent, except when the sample is a large proportion (more than 5%) of a finite population. Hence, if $X_1, X_2, ..., X_n$ are simple random samples, then $X_1, X_2, ..., X_n$ may be treated as independent random variables, all with the same distribution and is said that they are independent and identically distributed (i.i.d.). The sample mean, denoted as $\overline{X}$, can be treated as the linear combination of means of different samples and is given as:

Random Variables (Part 2: Continuous Random Variables)

Mon, 12 Nov 2018 03:01:07 +0100

#### Continuous Random Variables :

A continuous random variable is a random variable which can take infinitely many values. The probabilities associated with a continuous RV is defined by probability density function(PDF).

#### Probability Density Function (PDF) :

As a continuous RV takes infinite values, the probability $P(X=x)$ for it can not be defined and takes a value of 0. Instead we define a probability density funaction, which intutively depicts probability per unit space, where space is defined by the range of the underlying random variable. For a continuous RV $X$ with a PDF $f(x)$, the probability can be given as:

Random Variables (Part 1: Discrete Random Variables)

Sun, 11 Nov 2018 13:11:27 +0100

Random Variable is a variable whose possible values are numerical outcomes of a random phenomenon or experiment. In other words, a random variable assigns a numeric value to each outcome of a random experiment. For example, if we roll a fair die, the random variable $X$ describing the experiment will take the values $1,2,3,4,5,6$ (which are the possible outcomes of the experiment). There are two types of random variables: discrete and continuous.
#### Discrete Random Variable :

A random variable is discrete if its possible values form a discrete set. The random variable denoting the experiment of rolling a fair die is an example of discrete random variable.