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Rutgers - STAT 960 - Class Notes - Week 7

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Week 7: Probability Distributions

18 October 2016

Basic Statistics for Research

Professor HK Dong

Wendy Liu

Baye’s Rule – given k mutually exclusive & exhaustive events such that and an observed event A, then

∙ Permits revising old probabilities based on new info

o Prior probability – associated w/ B and before we know status of A

▪ and

o Posterior probability – updated probability of B

▪

∙ Application of conditional probability

∙ Mutually exclusive events

Intro to probability distributions

Random variable – represents a possible numerical outcome from an uncertain event

Probability distribution – list of the distinct numerical values of discrete random variable X along with their associated probabilities

gives the probability for each value and satisfies:

1. for each value of X

2.

Summary measures

∙ Expected value – mean of probability distributions; weighted average of all possible values o

∙ Variance – weighted average of squared deviations about the mean

o

o ----- alternative formula Don't forget about the age old question of How do you organize your product for maximum productivity?

We also discuss several other topics like What are the top two gases in the atmosphere, how much is there of each?

∙ Standard deviation – positive square root of variance

o

Probability Distributions

1. Discrete

a. Binomial

b. Poisson

c. Hypergeometric (we’re not responsible for knowing this type)

2. Continuous

a. Normal

b. Uniform

c. Exponential

Binomial probability distribution’s 3 critical properties

1. Result of each trial may be failure or success

2. Probability of success is same for each trial

3. Trials are independent If you want to learn more check out What is the meaning of aerobic in respiration?

Notation for binomial distributions

S success

F failure

n # of trials

x specific # of successes (0 ≤ x ≤ n)

p probability of successes in one trial

(1-p) probability of failure in one trial

P(x) probability of getting exactly x successes among n trials

Calculating P(x)

∙ Direct listing

o Count # of ways each outcome can occur

∙ General formula

o

Binomial Distribution with n trials and success probability p:

∙ Mean

o

∙ Variance

o

∙ Standard deviation

o

Poisson distribution – describes the # of times an event S can occur in an interval (time or space)

Poisson distribution requirements

∙ Random variable X is the # of occurrences of an S over some interval If you want to learn more check out What does eliot noyes known for?

∙ Occurrences must be random

∙ Independence: # of times S occurs in any interval is independent of the # of occurrences in another interval

∙ Lack of clustering: the chance of 2+ occurrences happening simultaneously is negligible and can be assumed to be 0

∙ Rate: the avg. # of occurrences per unit time is a constant λ, which does not change over time

If you want to learn more check out When does the united states declare war on the british?

x # of events in an area of opportunity λ expected # of events

e base of natural log (2.718……)

mean

variance

Poisson Approximation for Binomial Distributions We also discuss several other topics like When are the differences in kinds of questions for linguistic anthropologists vs. linguists?

∙ Binomial distribution w/ large n (n≥100)

∙ Very small p (np≤10)

∙ np = λ of moderate magnitude