STAT 164 - Probability Distributions SUMMARY CHAPTER 2

• BERNOULLI

• BINOMIAL

• GEOMETRIC

• NEGATIVE BINOMIAL

• HYPERGEOMETRIC

• POISSON

• MULTINOMIAL

What are the types of DISCRETE DISTRIBUTIONS?

BERNOULLI

RANDOM EXPERIMENT

One trial with only two outcomes: success or failure

BERNOULLI

RANDOM EXPERIMENT

The probability of success is equal to p and the probability of failure is 1 - p

BERNOULLI

RANDOM VARIABLE (X)

Defined as 1 if a trial results in success and 0 if the same trial results in failure

PROBABILITY FUNCTION

BERNOULLI

MEAN

BERNOULLI

VARIANCE

BERNOULLI

DISTRIBUTION

BINOMIAL

BINOMIAL

RANDOM EXPERIMENT

The n identical Bernoulli trials are conducted independently, that is, the outcome of any one trial does not depend on the outcome of the other trial.

BINOMIAL

GEOMETRIC

RANDOM EXPERIMENT

The probability of success does not change from trial to trial

BINOMIAL

RANDOM EXPERIMENT

Sampling is with replacement

BINOMIAL

RANDOM VARIABLE (X)

Number of success/failures in n trials

PROBABILITY FUNCTION

BINOMIAL

MEAN

BINOMIAL

VARIANCE

BINOMIAL

DISTRIBUTION

GEOMETRIC

GEOMETRIC
HYPERGEOMETRIC

RANDOM EXPERIMENT

Each trial has two possible outcomes: success or failure

GEOMETRIC

RANDOM EXPERIMENT

The trials are independent.

GEOMETRIC

RANDOM VARIABLE (X)

• Number of trials to get the first success

• Number of trials BEFORE observing the first success

PROBABILITY FUNCTION, MEAN, and VARIANCE

given x = 1, 2, …….

GEOMETRIC

PROBABILITY FUNCTION, MEAN, and VARIANCE

given x = 0, 1, 2, …….

GEOMETRIC

DISTRIBUTION

NEGATIVE BINOMIAL

NEGATIVE BINOMIAL

RANDOM EXPERIMENT

A generalization of the geometric distribution

NEGATIVE BINOMIAL

RANDOM VARIABLE (X)

Number of trials to get r successes

PROBABILITY FUNCTION

NEGATIVE BINOMIAL

MEAN

NEGATIVE BINOMIAL

VARIANCE

NEGATIVE BINOMIAL

DISTRIBUTION

HYPERGEOMETRIC

HYPERGEOMETRIC

RANDOM EXPERIMENT

Sampling is without replacement

HYPERGEOMETRIC

RANDOM VARIABLE (X)

• Number of success/failures in n draws

OR

• Number of success/failures in a sample of size n

where N is the total number of objects, D is the number of success from N, and n is the number of objects drawn without replacement.

PROBABILITY FUNCTION

HYPERGEOMETRIC

MEAN

HYPERGEOMETRIC

VARIANCE

HYPERGEOMETRIC

DISTRIBUTION

POISSON

POISSON

RANDOM EXPERIMENT

outcomes are discrete

POISSON

RANDOM EXPERIMENT

the number of successes, k, in any interval is independent of the number of successes in any other interval

POISSON

RANDOM EXPERIMENT

the chance of success is extremely small

POISSON

RANDOM VARIABLE (X)

Number of success/failure/events in a unit of time or space or interval

PROBABILITY FUNCTION

POISSON

MEAN

POISSON

VARIANCE

POISSON

MULTINOMIAL

RANDOM EXPERIMENT

an event with k different outcomes, each of which is observed n_i times in N trials

MULTINOMIAL

RANDOM VARIABLE (X)

(one random variable per outcome) e.g. 1st outcome X₁= number of 1^st outcome in n trials

PROBABILITY FUNCTION

MULTINOMIAL

MEAN

MULTINOMIAL

VARIANCE

MULTINOMIAL

• NORMAL

• STANDARD NORMAL

• EXPONENTIAL

What are the types of CONTINUOUS DISTRIBUTIONS?

DISTRIBUTION

NORMAL

NORMAL

Its curve is bell-shaped and unimodal

NORMAL

Mean = Median = Mode

NORMAL

Symmetric about the mean

NORMAL

50% of the values are less than the mean and 50% are greater than the mean

NORMAL

Empirical Distribution Theorem

• Approximately 68% of the data falls within one standard deviation from the mean

• Approximately 95% of the data falls within two standard deviation from the mean

• Approximately 99.7% of the data falls within three standard deviation from the mean

PROBABILITY FUNCTION

NORMAL

MEAN

NORMAL

VARIANCE

NORMAL

DISTRIBUTION

STANDARD NORMAL

PROBABILITY FUNCTION

STANDARD NORMAL

MEAN

STANDARD NORMAL

VARIANCE

STANDARD NORMAL

λ = 1/λ

DISTRIBUTION

EXPONENTIAL

EXPONENTIAL

The continuous counterpart of the geometric distribution

EXPONENTIAL

The only parameter is the failure rate, λ which is constant

EXPONENTIAL

Appropriate for modelling life length data, survival time, or time between Poisson events

EXPONENTIAL

The probability decreases over time at a constant rate

PROBABILITY FUNCTION

EXPONENTIAL

MEAN

EXPONENTIAL

VARIANCE

EXPONENTIAL

The cumulative distribution function of an EXPONENTIAL distribution is given by: