#
Random Variables:

Mean, Variance and

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

### Example: Tossing a single **unfair** die

For fun, imagine a **weighted** die (cheating!) so we have these probabilities:

1 | 2 | 3 | 4 | 5 | 6 |

0.1 |
0.1 |
0.1 |
0.1 |
0.1 |
0.5 |

### Mean or Expected Value: μ

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

### Example continued:

x | 1 | 2 | 3 | 4 | 5 | 6 |

p | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.5 |

xp | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 3 |

μ = Σxp = 0.1+0.2+0.3+0.4+0.5+3 = 4.5

The expected value is 4.5

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

### Variance: Var(X)

The Variance is:

Var(X) = Σx^{2}p − μ^{2}

To calculate the Variance:

- square each value and multiply by its probability
- sum them up and we get
**Σx**^{2}p - then subtract the square of the Expected Value
**μ**^{2}

### Example continued:

x | 1 | 2 | 3 | 4 | 5 | 6 |

p | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.5 |

x^{2}p |
0.1 | 0.4 | 0.9 | 1.6 | 2.5 | 18 |

Σx^{2}p = 0.1+0.4+0.9+1.6+2.5+18 = 23.5

Var(X) = Σx^{2}p − μ^{2} = 23.5 - 4.5^{2} = 3.25

The variance is 3.25

### Standard Deviation: σ

The Standard Deviation is the square root of the Variance:

σ = √Var(X)

### Example continued:

x | 1 | 2 | 3 | 4 | 5 | 6 |

p | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.5 |

x^{2}p |
0.1 | 0.4 | 0.9 | 1.6 | 2.5 | 18 |

σ = √Var(X) = √3.25 = 1.803...

The Standard Deviation is 1.803...

Let's have another example!

(Note that we run the table downwards instead of along this time.)

### 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 **x ^{2}p**:

Probability p |
Earnings ($'000s) x |
xp |
x ^{2}p |
---|---|---|---|

0.2 | -50 | -10 | 500 |

0.3 | 0 | 0 | 0 |

0.4 | 50 | 20 | 1000 |

0.1 | 150 | 15 | 2250 |

Σp = 1 | Σxp = 25 | Σx^{2}p = 3750 |

μ = Σxp = **25**

Var(X) = Σx^{2}p − μ^{2}

= 3750 − 25^{2}

= 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 |
x ^{2}p |
---|---|---|---|

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 | Σx^{2}p = 4250 |

μ = Σxp = **45**

Var(X) = Σx^{2}p − μ^{2}

= 4250 − 45^{2}

= 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) = Σx
^{2}p − μ^{2} - The
**Standard Deviation**is: σ = √Var(X)