Random Variables - Mean, Variance, Standard Deviation

A Random Variable is a set of possible values from a random experiment.

Example: Tossing a coin: we could get Heads or Tails.

Let's give them the values Heads=0 and Tails=1 and we have a Random Variable "X":

 

So:

  • We have an experiment (like tossing a coin)
  • We give values to each event
  • The set of values is a Random Variable

Learn more at Random Variables.

Mean, Variance and Standard Deviation

They have special notation:

  • μ is the Mean of X and is also called the Expected Value of X
  • Var(X) is the Variance of X
  • σ is the Standard Deviation of X

Mean or Expected Value

When we know the probability p of every value x we can calculate the Expected Value (Mean) of X:

μ = Σxp

Note: Σ is Sigma Notation, and means to sum up.

To calculate the Expected Value:

  • multiply each value by its probability
  • sum them up

It is a weighted mean: values with higher probability have higher contribution to the mean.

 

Variance

The Variance is:

Var(X) = Σx2p − μ2

To calculate the Variance:

  • square each value and multiply by its probability
  • sum them up and we get Σx2p
  • then subtract the square of the Expected Value μ2

 

Standard Deviation

The Standard Deviation is the square root of the Variance:

σ = √Var(X)

 

An example will help!

 

You plan to open a new McDougals Fried Chicken, and found these stats for similar restaurants:

Percent Year's Earnings
20% $50,000 Loss
30% $0
40% $50,000 Profit
10% $150,000 Profit

Using that as probabilities for your new restaurant's profit, what is the Expected Value and Standard Deviation?

 

The Random Variable is X = 'possible profit'.

Sum up xp and x2p:

Probability
p
Earnings ($'000s)
x

xp

x2p
0.2 -50 -10 500
0.3 0 0 0
0.4 50 20 1000
0.1 150 15 2250
Σp = 1   Σxp = 25 Σx2p = 3750

 

μ = Σxp = 25

Var(X) = Σx2p − μ2 = 3750 − 252 = 3750 − 625 = 3125

σ = √3125 = 56 (to nearest whole number)

But remember these are in thousands of dollars, so:

  • μ = $25,000
  • σ = $56,000

So you might expect to make $25,000, but with a very wide deviation possible.

Let's try that again, but with a much higher probability for $50,000:

Example (continued):

Now with different probabilities (the $50,000 value has a high probability of 0.7 now):

Probability
p
Earnings ($'000s)
x

xp

x2p
0.1 -50 -5 250
0.1 0 0 0
0.7 50 35 1750
0.1 150 15 2250
Σp = 1 Sums: Σxp = 45 Σx2p = 4250

 

μ = Σxp = 45

Var(X) = Σx2p − μ2 = 4250 − 452 = 4250 − 2025 = 2225

σ = √2225 = 47 (to nearest whole number)

In thousands of dollars:

  • μ = $45,000
  • σ = $47,000

The mean is now much closer to the most probable value.

And the standard deviation is a little smaller (showing that the values are more central.)

 

Continuous

Random Variables can be either Discrete or Continuous:

  • Discrete Data can only take certain values (such as 1,2,3,4,5)
  • Continuous Data can take any value within a range (such as a person's height)

Here we looked only at discrete data, as finding the Mean, Variance and Standard Deviation of continuous data needs Integration.

 

Summary

A Random Variable is a variable whose possible values are numerical outcomes of a random experiment.

The Mean (Expected Value) is: μ = Σxp

The Variance is: Var(X) = Σx2p − μ2

The Standard Deviation is: σ = √Var(X)