W9 Discrete Probability Distribution

random variable

is a variable that takes on numerical values

determined by the outcome of a random experiment.

Random

It is called ___ because its value depends on chance.

Variable

 It is called a ___ because it can take different possible

values.

Discrete Random Variable

• Takes on a countable number of possible

values.

• countable outcomes (whole numbers).

• Values can be listed (e.g., 0, 1, 2, 3, ...)

• Number of students present in class, number of heads in coin tosses

Continuous Random Variable

• Takes on an uncountable number of values within an interval.

• measurable outcomes (decimals possible).

• Values can take any value within a range (fractions, decimals)

• Height of students, time taken to finish a race, temperature

probability distribution

A  ___ of a discrete random variable shows

how the total probability (which is 1) is distributed among all

possible values of the variable.

• It lists each possible value of the random variable along with

its corresponding probability.

discrete probability distribution

It must satisfy these two main

properties:

•  Each probability is between 0 and 1

•  The sum of all probabilities equals 1

•  Each probability corresponds to a possible value of the random

variable.

Probability Mass Function

• gives the probability that discrete variable takes on a specific value 

Valid Probability Distribution

• All probabilities must be non-negative.

• The sum of all probabilities must equal 1.

• If these are not met, the table or PMF is invalid.

Expected Value (Mean) of a Discrete Random Variable

• Reports the central location of the data

•  The Expected Value or Mean of a discrete random variable gives the longrun average value of the variable after many repetitions of the experiment.

• It represents what you expect to happen on average.

Variance

it measures how much the values of the random

variable deviate from the mean.

Standard Deviation

is simply the square root of the variance, giving a measure of

spread in the same unit as X.

1. Expected Value

2. Variance

3. Standard Deviation

1. ___Long-term average outcome

2. ___ Measure of spread of probabilities

3. ___Square root of variance (same unit as

data)Real-life use: risk assessment, business forecasting, game

fairness, insurance pricing

1. Binomial Distribution

2. Poisson Distribution

3. Geometric Distribution

4. Hypergeometric distribution

Discrete probability distributions describe situations where a

random variable takes countable values (0, 1, 2, 3, ...).

• Below are the four most common types used in statistics and

probability.

Binomial Distribution

Characteristics

• The experiment consists of n independent trials.

• Each trial has two possible outcomes: success or failure.

— Customer can purchase or not purchase

— Citizen can vote or not vote

• The probability of success, p, is the same for each trial.

• The random variable X = number of successes in ntrials.

• The random variable is the result of counting the number of successes (this will be the horizontal axis)

•  The probability of success must be the same for each trial (this is not the probability we will be calculating)

Binomial Distribution Has The Following

Characteristics:

• Each trail must be independent of any other trial

• The outcome of one trial does not affect the outcome of any other

trial

• There is no pattern

* Example: Multiple choice exam: a,a,a, b,b,b, a,a,a, ,b,b,b...

Binomial Distribution Has The Following

Characteristics: