Module 4: Discrete Probability Distribution

Random Variable

A numerical outcome of a random experiment

Discrete Random Variable

A random variable that can take up a finite number of outcomes

Probability Distribution

A table, graph or formula used to describe the probability of each value of a discrete random variable

Properties of a Probability Distribution

1. All of the probabilities add up to 1

2. Each P(x) is a value between 0 and 1

Probability Mass Function

• asks “what is the probability of getting exactly that outcome/number?”

• found within each individual rows of a table that stores all the probabilities of above said question

Cumulative Probability Function

• asks “what is the probability of getting something less than or equal to a certain number:

Expected Value

• The long-term average/mean (µ): the value you will slowly expect to get as you keep doing an experiment over and over again

• Formula (Sum of all the products of X times each of its own Probability)

Variance

• measure of spread of values around the mean

• Step 1, x - mean/expect value for every row

• Step 2, square each of the rows after doing x - mean/expected value to remove any negatives

• Step 3, multiply whatever you get after square by the P" (on each of its found rows)

• Step 4, add em all up together (Sum)

Standard Deviation

• Square root of variance

• When finding it, just do the variance formula then just square root the whole thing

Binomial Distribution

Probability distribution that describes number of successes in a fixed number of independent trials

Number of trials symbol (Binomial Distribution)

n

Probability of success symbol (Binomial Distribution)

p

Probability of failure symbol and formula (Binomial Distribution)

• q

• q = 1-p

Bernoulli Trial

Single experiement/trial with only two outcomes: Success / Failure (p/q)

Expected value/mean formula (Binomial Distribution)

Average number of successes in n trials: n x p

Variance (Binomial Distribution)

• n x p x q

Standard Deviation

(square root of npq - the variance formula)

Geometric Trials

Number of trials needed in order to achieve the first success in a sequence of independent Bernoulli trials

Poisson Distribution

The number of occurrences of an event in a fixed time/space, given that they are independent and occur at a constant rate

Lambda (λ)

expected/average number of occurrences in a given time

Euler’s number

2.718

What happens if the time period for the question in lambda is different

Multiply lambda by the new time period relative to the old one. ex. if 1 hour lambda 6, then 30 minutes is ½ hours x lambda 6 which is 3 (the new lambda)