Discrete Probability Distributions

Random Variable

Value that represents the outcome of a trial/ experiment/ event

• functions or rules that assign numerical values to each outcome in the experiment’s sample space

• X = random variable in general

• X_i = specific value of X

Probability distribution function

Shows probabilities of different possible value of a variable

For discrete, it shows the probability of each possible value of the variable

Describes likelihood of obtaining the possible values

Law of probability

Probability that I choose each individual item (or probability of each outcome) must equal 1

Cumulative distribution function (CDF)

Shows cumulative sum of probabilities from smallest to largest possible values of X

Can’t go over 1, constantly increasing until 1 and tapers off after

Expected value of random variable X =

Weighted average

E(X) is sum of all possible values of X weighted by respective probabilities

mean of probability distribution

Binomial distribution

Used with exactly 2 mutually exclusive outcomes

Dichotomous → true/ false, yes/ no, etc.

Bimodal characteristics

• fixed number of trials n

• only two possible outcomes

• probability of success for each trial remains constant

• trials are independent from each other

• random variable X represents number of successes out of n trials

How to calculate binomial PDF in Excel

=BINOM.DIST ( )

cumulative probability = ‘TRUE’

Poisson Distribution

Shows how many times an event is likely to occur/ arrive over ap eriod of time

• random variable X is the number of events/ arrivals per unit of time

• parameter is (wishbone symbol), is mean number of events per unit of time in space

Poisson characteristics

• event occurs randomly over time or space

• Average arrival rate (wishbone figure) remains constant

• arrivals independent of each other

• random variable X represents the number of arrivals within observed time interval

• always right-skewed → discrete but still infinite, would go on forever but never touch 0

Poisson distribution in Excel

=POISSON.DIST ( )

TRUE for cumulative function