Discrete Probability Distributions

A Random Variable “x” is determined by chance Or “could be” determined by chance? The important thing: it’s some value we get in a single trial of a probability experiment It’s what we’re measuring

Discrete vs. Continuous Discrete A countable number of values “Red”, “Yellow”, “Green” 2 of diamonds, 2 of hearts, … etc. 1, 2, 3, 4, 5, 6 rolled on a die Continuous All real numbers in some interval An age between 10 and 80 ( and ) A dollar amount A height or weight

Discrete is our focus for now Discrete A countable number of values (outcomes) “Red”, “Yellow”, “Green” 2 of diamonds, 2 of hearts, … etc. 1, 2, 3, 4, 5, 6 rolled on a die Continuous Will talk about continuous probability distributions in future chapters.

Start with a frequency distribution General layout A specific made-up example How many children live here? Number of households or more10 Total responses450 OutcomeCount of occurrences

Include a Relative Frequency column General layout A specific simple example # of child ren Number of households Relative Frequency Total OutcomeCount of occur- rences Relative Frequency =count ÷ total

You can drop the count column General layout A specific simple example # of childrenRelative Frequency Total1.000 OutcomeRelative Frequency =count ÷ total

Sum MUST BE EXACTLY 1 !!! In every Probability Distribution, the total of the probabilities must always, every time, without exception, be exactly – In some cases, it might be off a hair because of rounding, like for example. – If you can maintain exact fractions, this rounding problem won’t happen.

Answer Probability Questions What is the probability … …that a randomly selected household has exactly 3 children? …that a randomly selected household has children? … that a randomly selected household has fewer than 3 children? … no more than 3 children? A specific simple example # of childrenRelative Frequency Total1.000

Theoretical Probabilities Rolling one die ValueProbability 11/ Total1 Total of rolling two dice ValueProb.ValueProb. 21/3685/36 32/3694/36 43/36103/36 54/36112/36 65/36121/36 76/36Total1

Tossing coin and counting Heads One Coin How many headsProbability 01 / 2 1 Total1 Four Coins How many headsProbability 01/16 14/16 26/16 34/16 41/16 Total1

Tossing coin and counting Heads How did we get this?Four Coins How many headsProbability 01/16 13/16 26/16 33/16 41/16 Total1 Could try to list the entire sample space: TTTT, TTTH, TTHT, TTHH, THTT, etc. Could use a tree diagram to get the sample space. Could use nCr combinations. We will formally study The Binomial Distribution soon.

Graphical Representation Histogram, for exampleFour Coins How many headsProbability 01/16 14/16 26/16 34/16 41/16 Total1 6/16 4/16 1/ heads Probability

Shape of the distribution Histogram, for exampleDistribution shapes matter! 6/16 3/16 1/ heads Probability This one is a bell-shaped distribution Rolling a single die: its graph is a uniform distribution Other distribution shapes can happen, too