# S-Cool Revision Summary

## S-Cool Revision Summary

A random variable is a variable which takes numerical values and whose value depends on the outcome of an experiment. It is discrete if it can only take certain values.

For a discrete random variable X with **P(X = x)** then probabilities always sum to 1.

**P(X = x) = 1** (Remember means 'sum of').

'Cumulative' gives us a kind of running total, so a cumulative distribution function gives us a running total of probabilities within our probability table.

The cumulative distribution function, **F(x)** of X is defined as:

The expectation is the expected value of X, written as E(X) or sometimes as .

The expectation is what you would expect to get if you were to carry out the experiment a large number of times and calculate the 'mean'.

To calculate the expectation we can use: E(X) = x P(X = x)

If X is a discrete random variable and f(x) is any function of x, then the expected value of f(x) is given by: E[f(x)] = f(x)P(X = x)

There are a few general results we should remember to help with our calculations of expectations: E(a) = a

E(aX + b) = aE(X) + b

The variance is a measure of how spread out the values of X would be if the experiment leading to X were repeated a number of times.

The variance of X, written as Var(X) is given by:

**Var(X) = E(X ^{2}) - (E(X))^{2}**, If we write E(X) = then,

**Var(X) = E(X ^{2}) - ^{2}**

Or **Var(X) = E(X - ) ^{2}**, this tells us that Var(X) 0

There are a few general results we should remember to help with our calculations of variances:

Var(aX) = a^{2}Var(X)

Var(aX + b) = a^{2}Var(X)

The square root of the Variance is called the Standard Deviation of X. Standard deviation is given the symbol .

= or Var(X) = ^{2}

Suppose that an experiment consists of n identical and independent trials, where for each trial there are 2 outcomes.

'Success' which is given probability **p**

'Failure' which is given probability **q** where **q = 1 - p**

Then if X = the number of successes, we say that X has a binomial distribution. We write:

Sometimes written as: **X ~ bin(n, p)**

If our random variable follows a binomial distribution then the associated probabilities are calculated using the following formula:

gives us the number of ways of choosing r objects from **n**.

It is calculated by:

You may also have a button on your calculator that will do all that for you.

If: X ~ bin(n, p)

Then remember:

E(X) = n x p

Var(X) = n x p xq

**n** = number of trials

**p** = probability of a success

**q** = probability of a failure = 1 - p

If we let X be the random variable of the number of trials up to and including the first success then X has a Geometric distribution.

The probabilities are worked out like this:

(Remember, **p** = probability of success, and **q** = probability of failure)

P(X = 1) = p |
No failure, success on first attempt (hooray!) |

P(X = 2) = q x p |
1 failure then success |

P(X = 3) = q |
2 failures then success |

P(X = 4) = q |
3 failures then success |

P(X = r) = q |
r - 1 failures followed by success |

If X follows a Geometric distribution we write: **X ~ Geo(p)**

This reads as 'X has a geometric distribution with probability of success, p'.

A special result to remember: P(X > r) = q^{r}

If: X ~ Geo(p)

Then:

E(X) = and Var(X) =