Statistics Probability And Randomness

Front 1

random phenomenon

Back 1

individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions

Front 2

probability

Back 2

proportion of times the outcome would occur in a long series of repetitions

Front 3

independent trials

Back 3

the outcome of one trial must not influence the outcome of any other

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Sample space S (random phenomenon)

Back 4

The set of all possible outcomes

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Event

Back 5

an outcome or a set of outcomes of a random phenomenon (a subset of the sample space)

Front 6

4 Probability rules

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1. 0 ≤ P(A) ≤ 1
2. P(S) = 1
3. Disjoint events (no outcomes in common)-P(A or B) = P(A) + P(B)
4. Complement rule P(Ac) = 1 - P(A)

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Disjoint events

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Events that have no outcomes in common

Front 8

Probability of any event (description)

Back 8

Equals the sum of the probabilities of the outcomes making up the event

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Probability of any event A (equation)

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P(A) = (count of outcomes in A) / (count of outcomes in S)

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Multiplication rule for independent events

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Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs so P(A and B) = P(A)P(B)

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Disjoint and independet

Back 11

Disjoint events cannot be independent (they can't occur together, so the fact that A occurs already says that B cannot occur)

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Random variable

Back 12

A variable whose value is a numerical outcome of a random phenomenon

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Discrete random variable (X)

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A random variable with a finite number of possible values

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Probability distribution of discrete X

Back 14

Lists the values and their probabilities

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Probabilities of values of X (must satisfy these two things)

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1. Every probability pi is a number between 0 and 1
2. p1 + p2 + ... + pk = 1
(probability of any event comprising any of the xi values is found by adding up their respective pi values)

Front 16

Probability histograms

Back 16

Compare the probability of certain numerical outcomes

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Continuous random variable X

Back 17

Takes all values in an interval of numbers

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Probability distribution of continuous X

Back 18

Described by a density curve; the probability of any event is the area under the density curve and above the values of X that make up the event

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Continuous probability distributions and individual outcomes

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All are assigned probabilities of zero

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Continuous probaility of normal distribution

Back 20

Calculate probabilities using z-scores

Z = (X - µ) / σ

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Mean of discrete random variable X

Back 21

Σxipi = µx = x1p1 + x2p2 + ... + xnpn

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Law of large numbers

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Draw independent observations at random from any population with finite mean µ. Decide how accurately you would like to estimate µ. As the number of observations drawn increases, the mean x-bar of the observed values eventually approaches the mean µ of the population as closely as you specified and then stays that close.

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Variability of the random outcomes

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The more variable the outcomes, the more trials are needed to ensure that mean outcome x-bar close to the distribution mean µ

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Linear transformation of the mean of a random variable

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µ(new) = a + bµ

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Addition of random variable means

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µ(x+y) = µ(x) +µ(y)

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Variance of a discrete random variable

Back 26

Σ(xi - µx)^2(pi)

(standard dev, is sqrt)

Front 27

Independence and adding variance

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Variances only add when there is no association between variables

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Variance of the sum of two non-independent variables

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Depends on the correlation between the variables (it is non-zero whenever two random variables are not independent)

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Linear transformation (a, b) for variances

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σ^2 (new) = b^2σ^2

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σ^2(X+Y) or σ^2(X-Y), independent equals

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σ^2(X) +σ^2(Y)

Front 31

Adding variances with correlation p

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(X+Y)-->add 2pσ(X)σ(Y)
(X-Y)-->sub. 2pσ(X)σ(Y)

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Union (of collection of events

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The event that at least one of the collection occurs

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P(one or more of A, B, C)

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P(A) + P(B) + P(C)

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P(A or B) equals

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P(A) + P(B) - P(A and B)

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Conditional probability

Back 35

Computing probability given the fact that some event has already happened

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Multiplication rule: P(A and B) equals

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P(A)P(B l A)

(P(B) given A)

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Conditional probability (P(A)>0)

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P(B l A) = P(A and B) / P(A)

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Intersection

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Event that all of the events occur (out of any collection of events)

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P(A and B and C)

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P(A)P(B l A)P(C l A and B)

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Tree diagram

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(A and B)-->multiply each nodal probability along a branch
P(B l A)-->calculate probability starting from the node where A is decided

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Independent events (conditional probability)

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P(B l A) = P(B)

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Baye's Rule

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A1, A2, ..., Ak are disjoint events without P=0 and add to 1

P(Ai l C) = [P(C l Ai)P(Ai)] / [P(C l A1)P(A1) + ... + P(Ak)P(C l Ak)

C is another event whose probability is not 0 or 1