Probability Random Variables Preparatory Notes

of Probability

Definition
Probability Mass Function
Mean and Variance
Conditional Expectation
Poisson Distribution

Definition
Probability Density Function
Probability as an Integral
Mean and Variance
Conditional Expectation
Exponential Distribution
Normal Distribution

is uncertain (roll a die)
Sample Space: set of all possible outcomes of an experiment (S = {1, 2, 3, 4, 5, 6})
Event: a subset of outcomes that is of interest to us (e.g. event 1 = even number, event 2 = higher than 4, event 3 = throw a 2, etc)
Probability: measure of how likely an event is to occur (between 0 and 1)

to occur (between 0 and 1)

Classical Definition
- Calculate, assuming events equally likely
Relative Frequency Approach
- Doing an experiment, using historical data
Subjective/Bayesian Probability
- Trusting instinct, using judgement

inside parliament, 1000
politicians were interviewed, 600 of which were from the Yellow Party
and 400 were from the Blue party. Results of the survey are as follows:

LAWS & NOTATION:

chance of selection) from the survey is a member of the Yellow Party ( Y ):
P(Y) = 600/1000 = 0.6
Probability that a person chosen randomly from the survey would ban smoking ( Ban ):
P(Ban) = 500/1000 = 0.5

‘Basic’ Events

member
of the Yellow Party ( Y ) and would ban smoking ( Ban ), which is denoted by:

‘Combined’ Events 1
(Intersection of events)

Probability that a person chosen randomly from the survey would ban smoking ( Ban) and is a member of the Yellow Party ( Y ), denoted by:

Clearly the intersection of events does not depend on their order of listing.

( Y )
or would ban smoking ( Ban ):

‘Combined’ Events 2
(Union of events)

So here we have seen that:

This is an example of the Addition Law for Probabilities, see next slide …………….

A and B are mutually exclusive , i.e. they cannot both happen, in which case the addition law simplifies to:

( Ban ) given that he/she is from the Yellow Party ( Y ):

‘Combined’ Events 3
(Conditional events)

i.e.:

This is an example of the Multiplication Law for Probabilities, see next slide …………….

Or by simple rearrangement:

A and B are independent, i.e. the occurrence of A has no influence on the probability of B [and vice versa)]
i.e. P(B/A) = P(B) [and P(A/B)=P(A)]
in which case the multiplication law simplifies to:
NOTE: This notion of independence can be extended to more than 2 events.

to ban smoking inside parliament
could be reported as:
P(Y) = 0.6, P(Ban/Y) = 5/12
P(B) = 0.4 , P(Ban/B) = 5/8
Now P(Ban) = P(Ban∩Y) + P(Ban∩B) [as Ban∩Y and Ban∩B are mutually exclusive & the only two ways that Ban can occur]
= P(Ban/Y) x P(Y) + P(Ban/B) x P(B) [by multiplication rule]
= 5/12 x 0.6 + 5/8 x 0.4
= 0.25 + 0.25 = 0.5

This is an example of the Law of Total Probability, see next slide …………….

An are mutually exclusive and complete (i.e. one of them must occur), then the probability of another event (B) can be calculated by weighting the conditional probabilities of B, i.e:
P(B) = P(B/A1) x P(A1) + P(B/A2) x P(A2) + …P(B/An) x P(An)
See another example on next slide …..

of chocolates is packed into boxes on four production lines A, B, C, D. Records show that a small percentage of boxes are not packed properly for sale as follows

What is the probability that a box chosen at random from the factory’s output is faulty?

outcome of an experiment.
When the values are ‘discrete’, (e.g. value shown when die is thrown, first number drawn in lottery, daily number of patients attending A&E in Lancaster), the random variable is discrete.
{ For Later ***When the values are ‘continuous’ , (e.g. height of randomly selected student, temperature in Lancaster, time between patients arriving at A&E), the random variable is continuous.}

a discrete random variable (X) simply lists the probabilities of each possible value, x, of the random variable.

E.g. Sum of values when two dice are thrown:

mean, of a discrete random variable is defined as:

The variance of a discrete random variable is a measure of variability and is defined as:

N.B. σ is referred to as the standard deviation of X.

and Y are random variables and a and b are constants:
E(aX + bY) = aE(X) + bE(Y);
If X and Y are also independent random variables (i.e. the value of X has no influence on the value of Y (e.g. two throws of a die))
E(XY) = E(X)E(Y)
and
Var (aX + bY) = a2 Var(X) + c2 Var(Y).

variable

Suppose events A1, A2, A3, … An are mutually exclusive and complete (i.e. one of them must occur), then the expected value of a r.v. X can be calculated by weighting the conditional expected values of X, i.e.:
E(X) = E(X/A1) . P(A1) + E(X/A2) . P(A2) + …E(X/An) . P(An)

For example if 20% of the working population work at home and the remaining 80% travel an average of 10 miles to work, the overall average distance travelled to work is:
E(travel to work distance) = 0 x 0.2 + 10 x 0.8 = 8 miles.

modelling is the Poisson random variable.

and its variance σ2 = β.

And Poisson probabilities can be obtained directly from the formula, statistical tables or computer software.

A Poisson distribution with mean β has pmf:

λ per unit time;
No. of events in period of time T has a Poisson distribution with mean λT
Events in real stochastic processes (e.g. arrivals of customers at a bank, calls to a call centre, patients to an A&E department, breakdowns in equipment) occur ‘at random’ when there are a large number of potential customers/ callers/ patients/ components each with independent probabilities of arriving/ calling/ falling ill/ breaking.

Poisson Random Variable

an average rate of 0.6 per minute, then:
No. of arrivals in 5 minute period has Poisson distribution with mean = 3.

Poisson Random Variable

outcome of an experiment.
When the values are ‘continuous’ , (e.g. height of randomly selected student, temperature in Lancaster, time between patients arriving at A&E), the random variable is continuous.

X is captured by the probability density function (p.d.f.) f (x), which is as follows:
f (x) is never negative
the total area under f (x) is one (total probability)

pdf, i.e. f (x), for the appropriate range of values − e.g.

Methods to find areas under pdf are integration, statistical tables and computer software

mean, of a continuous random variable is defined as:

The variance of a continuous random variable is a measure of variability and is defined as:

N.B. σ is referred to as the standard deviation of X.

same as for discrete r.v.

If X and Y are random variables and a and b are constants:
E(aX + bY) = aE(X) + bE(Y);
If X and Y are also independent random variables (i.e. the value of X has no influence on the value of Y (e.g. two throws of a die))
E(XY) = E(X)E(Y)
and
Var (aX + bY) = a2 Var(X) + c2 Var(Y).

variable
NB. Exactly same as for discrete r.v.

Suppose events A1, A2, A3, … An are mutually exclusive and complete (i.e. one of them must occur), then the expected value of a r.v. X can be calculated by weighting the conditional expected values of X, i.e:
E(X) = E(X/A1) . P(A1) + E(X/A2) . P(A2) + …E(X/An) . P(An)

modelling is the Exponential random variable.

and its variance σ2 = (1/γ)2,
i.e. variance = mean2

E.g. f(x) for Exponential r.v. with mean=4.

And areas under curve follow from simple formula:

Exponential distribution with mean 1/γ has pdf:

λ per unit time (as is common in real stochastic processes – see earlier note);
The time between events has an Exponential distribution with mean 1/λ.
And the time to the next event has an Exponential distribution with mean 1/λ, whether or not an event has just occurred. [This is the memoryless property of the Exponential distribution – and is counter-intuitive!].
Conversely, if the gaps between events are independent and from an exponential distribution with mean 1/λ, the events occur ‘at random’ at rate λ per unit time.

Exponential Random Variable

time.
THEN: Time between events has an Exponential distribution, with mean of 1/A.
AND: Time to next event has an Exponential distribution, with mean of 1/A.

EXAMPLE
IF Arrivals to bank occur ‘at random’, at rate of 0.6 per minute.
THEN: Time between arrivals has an Exponential distribution, with mean = 1.667 minutes.
AND: Time to next arrival has an Exponential distribution, with mean of 1.667 minutes.

is the Normal random variable.

& hence its standard deviation is σ.

E.g. f(x) for Normal r.v. with mean=μ and stan dev σ..

And areas under curve are obtained from statistical tables or from computer software.

Point of inflection

μ

μ+σ

Normal distribution with mean μ and variance σ2 has pdf:

can be found by finding the ‘equivalent’ area under the standard Normal curve
(mean=0 & SD=1), i.e.: