Probability mass function

Question

Probability mass function

f(x) = 4x + 2 / 50

x = 0, 1, 2, 3, 4

Determine the mean and variance of the random variable. Rounded 2 decimal places.

Answer 1

I'm assuming that the probability mass function you have is f(x) = (4x+2)/50.

For mean of a random variable, this is also known as the "expected value". Think about this as a weighted sum of each value of the random variable against its probability. Definition of mean of a random variable is shown below:

E[X] = Σ(x*f(x))

For example, if you had a pdf that gave the following values for x = 0, 1, 2 respectively: 0.5, 0.1, 0.4, then the expected value in that case would be 0(0.5) + 1(0.1) + 2(0.4) = 0.9

You can try applying a similar method for your case. Try having a table with your "x" values mapping to the probability (ie: f(x))

For variance of a random variable, the definition is shown below:

Var[X] = Σ((x - E[X])² ). In statistics, Variance is simply a measure of how spread out the values are from the mean. Ie: values that are further from the average/center cause the variance to increase more!

A helpful trick for Var[X] to make your life a bit easier is this identity:

Var[X] = E[X^2] - (E[X])^2

Note that in my example shown for calculating E[X], E[X^2] would be equal to (0^2) * 0.5 + (1^2) * 0.1 + (2^2) * 0.4 = 1.7

You can apply a similar methodology for your use case in calculating E[X^2]. Once you have that along with your E[X] value, you can get Var[X], which is the variance of the random variable in your problem.

Feel free to let me know if this helps :)