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.
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
And strictly speaking,
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
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.
Generally speaking, variance is defined as
If is discrete:
If is continuous:
Using the properties of expectations, we can show .
The standard deviation is defined as
For two random variables and , the covariance is generally defined as
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 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.