University is starting for some students who took A’levels in 2016. And, one of my ex-students told me to share/ summarise the things to know for probability at University level. Hopefully this helps. H2 Further Mathematics Students will find some of these helpful.

**Random Variables**

Suppose is a random variable which can takes values .

is a discrete r.v. is is countable.

is the probability of a value of and is called the probability mass function.

is a continuous r.v. is is uncountable.

is the probability density function and can be thought of as the probability of a value .

**Probability Mass Function**

For a discrete r.v. the probability mass function (PMF) is

, where .

**Probability Density Function**

If

.

And strictly speaking,

.

Intuitively,

.

**Properties of Distributions**

For discrete r.v.

.

.

For continuous r.v.

.

.

**Cumulative Distribution Function**

For discrete r.v., the Cumulative Distribution Function (CDF) is

.

For continuous r.v., the CDF is

.

**Expected Value**

For a discrete r.v. X, the expected value is

.

For a continuous r.v. X, the expected value is

.

If , then

For a discrete r.v. X,

.

For a continuous r.v. X,

.

**Properties of Expectation**

For random variables and and constants , the expected value has the following properties (applicable to both discrete and continuous r.v.s)

Realisations of , denoted by , may be larger or smaller than ,

If you observed many realisations of , is roughly an average of the values you would observe.

**Variance**

Generally speaking, variance is defined as

If is discrete:

If is continuous:

Using the properties of expectations, we can show .

**Standard Deviation**

The standard deviation is defined as

**Covariance**

For two random variables and , the covariance is generally defined as

Note that

**Properties of Variance**

Given random variables and , and constants ,

This proof for the above can be done using definitions of expectations and variance.

**Properties of Covariance **

Given random variables and and constants

**Correlation**

Correlation is defined as

It is clear the .

The properties of correlations of sums of random variables follow from those of covariance and standard deviations above.