Random Variables Part 2 in Statistics

In the previous segment, you learnt how to construct the probability distribution of a discrete random variable, you will now learn how to calculate the mean, variance and standard deviation of a discrete random variable from Thomas in the upcoming video, with the help of an example.

In this video, you learnt how to calculate the mean, variance and standard deviation of a discrete random variable. Suppose the probability mass function of a random variable X is as shown in the table given below.

The mean of a random variable is also referred to as the expected value of the random variable. The expected value of a random variable X is the average value of X that we would  'expect' to get after performing the experiment an infinite number of times. As demonstrated in the video, the expected value does not even have to be a possible outcome of the trial.

$Mean\left(X\right)={\mu }_x=x_1\ast P(X=x_1)+x_2\ast P(X=x_2)+...+x_n\ast P(X=x_n)$

This can be understood using an example. Let’s say Y is a random variable that takes the value 0 if we get heads and 1 if we get tails on tossing a fair coin. As it is a fair coin, we expect Y to take the value 0 half the time and 1 half the time. Hence, the expected value of Y will be 0.5, although Y itself never takes the value 0.5.

The variance of X gives us an indication of the spread of the values from the mean that the random variable can take. The higher the variance, the more the values are spread out from the mean of the distribution.

$V\left(X\right)=P(X=x_1)\ast ({\mu }_x-x_1{)}^2+P(X=x_2)\ast ({\mu }_x-x_2{)}^2+...+P(X=x_n)\ast ({\mu }_x-x_n{)}^2$

The standard deviation is the most commonly used measure of the dispersion of the values of a random variable from its mean.

$Standarddeviation\left(X\right)={\sigma }_x=\sqrt{Variance\left(X\right)}$

So let's take an example to understand the formulas related to expected value better. Take an instance of rolling a fair die once. Each face of a die has 1/6 or 0.16 probability that it will be the face that will show up when the die is rolled. 

If we apply the expected formula, we get the expected value as 0.16(1) + 0.16(2) + ....+0.16(6) 

= 3.36

Variance would be as 0.16(3.36 -1)^2 + 0.16(3.36-2)^2 + ..... 0.16(3.36-6)^2

=  2.81 

And the Standard deviation as Squareroot of variance, which is 1.67. 

The upcoming segment will throw discuss continuous random variables.