## Discrete Random Variables

A variable is represented by a capital letter i.e.

This can be any particular

A discrete random variable can only be certain

**X**This can be any particular

**x value**A discrete random variable can only be certain

**discrete values****P(X = x)**is the probability that the random variable is the same as a particular value of x*You will be given P(X = x) in an exam*## Probability Distribution

A probability distribution is a table like above

**REMEMBER**that it's probability, so**ΣP****(X=x) = 1****What if:**

Here there are two different probabilities and a constant

So

So

**k + 3k + (2-1)k + (4-1)k = 1**

8k = 1

k = 1/88k = 1

k = 1/8

**What if:**

**P(1 < X < 5)**

This is the same as

**P(X = 2,3 or 4)**

P(X = 2) = 0.2

P(X = 3) = 0.3

P(X = 4) = 0.25

**P(1 < X < 5) = 0.75**

## Cumulative Distribution Function

## Mean / Expected Value and Variance

This gives the

**mean**of the data**E(X) = ΣxP(X=x)**

Using this formula means to multiply each P(X = x) by the corresponding x value and find the sum of these

This gives the mean of the data squared

This gives the mean of the data squared

To find the

**Variance**of the data

These can be translated:

You need to

You need to

**remember these**## Coding

During coding of bivariate data the variables are coded

E.g.

To

Rearrange to

This means

Using your

E(X) = 50 x 5.1 + 150

Finding the

E.g.

**Y =**with standard deviation of__X - 150__**2.5**and the variable Y mean being**5.1****50**To

**find E(X)**Rearrange to

**X = 50Y + 150**This means

**E(X) = E(50Y + 150)**Using your

**transformation laws****E(X) = 50E(Y) + 150**E(X) = 50 x 5.1 + 150

**E(X) = 406**Finding the

**variance**is similar## Modeling

When you have a

So when

Then

For this

**discrete uniform distribution**, where the**probability**for every X value is the**same**, you can create a model which allows you to**work out the mean and variance**of X values up to an unknown amount (**n**)So when

**x = 1,2,3... n**Then

**P(X=x) = 1/n**For this

**E(X) =**__n + 1__**2****Var(X) =**__(n + 1)(n - 1)__**12**