Chapter 2: Random Variables & Probability Distribution Flashcards by RAMIHA GENEVIEVE VARGAS

What is a random experiment?

A process of drawing observation capable of repetition under the same conditions with unpredictable outcomes per trial.

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A set or collection of all possible outcomes of a random experiment that may be finite or infinite which are aka outcomes or sample points.

Sample Space

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A subset of sample space which indicates the occurence of the event.

Event

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Numerical value ranging from 0 to 1 measuring the likelihood of an event to occur.

Probability

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A rule that assigns exactly one real number to every possible outcome.

Random Variable

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The list of all possible values of x along with their corresponding probability whose sum is equal to 1.

Probability Distribution

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Formula for expected value if discrete.

E[X] = sum of xP

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Formula for variance.

E[X raised to 2] = sum of x squared P

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Discrete Probability Distribution: where there are only 2 outcomes and conducted independently. Success = p, Failure = 1 - p.

Bernoulli

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Discrete Probability Distribution: where several Bernoulli trials/experiments are conducted and the number of trials is known in advance.

Binomial

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Notation for Binomial or Bernoulli.

X~B(n,p)

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Formula for Binomial or Bernoulli.

P[X=x] = (n taken x) p raised to x (1-p) raised to n-x

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Mean for Bernoulli or Binomial

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Variance for Binomial or Bernoulli.

np(1-p)

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Discrete Probability Distribution: x is defined as the # of trials required to get the first success.

Geometric Random Experiment

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Formula for Geometric Random Experiment.

P[X=x] = p(1-p) raised to x-1

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Notation for Geometric Random Experiment.

X~Geometric(p)

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Mean for Geometric Random Experiment.

1/p

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Variance for Geometric Random Experiment.

(1-p)/p squared

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What is Y in Geometric Random Experiment?

Number of trials before the first success.

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Discrete Probability Distribution: where x is the number of trials needed to get r successes and is an infinite random variable.

Negative Binomial Random Experiment

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Formula for Negative Binomial Random Experiment.

P[X=x] = (x-1 taken r-1) p raised to r (1-p) raised to x-r

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Notation for Negative Binomial Random Experiment.

X~NBin(r,p)

Mean for Negative Binomial Random Experiment.

r/p

Variance for Negative Binomial Random Experiment.

r(1-p) all over p squared

Discrete Probability Distribution: probability changes from 1 trial to another and is most applied in biology.

Hypergeometric Random Experiment

Formula for Hypergeometric Random Experiment.

P[X=x] = (D taken x)(n-d taken n-x) all over (N taken n)

Notation for Hypergeometric Random Experiment.

X~Hypergeom(D,N,n)

Mean for Hypergeometric Random Experiment.

nD all over N

Variance for Hypergeometric Random Experiment.

nD(N-D)(N-n) all over N

Discrete Probability Distribution: describes the number of times an event occurs in a unit of time or space where chance of success is extremely small and is infinite.

Poisson Random Experiment

Formula for Poisson Random Experiment.

P[X=x] = e raised to negative lambda times lambda raised to x all over x!

Mean and Variance of Poisson Random Experiment.

Lambda

Discrete Probability Distribution: more than 2 possible outcomes, independent of trials, and fixed probability.

Multinomial Random Experiment

Properties of a Normal Curve: Notation.

X~n(mean, variance)

Properties of a Normal Curve: percentage of values less and greater than the mean.

50%

Properties of a Normal Curve: Measures of Tendency

Mean = Mode = Median

Properties of a Normal Curve: Formula.

f(x) = 1 over sigma square root of 2pi times e raised to -1/2 (x-mean all over sigma) squared.

Properties of a Normal Curve: Total area under the curve.

Properties of a Normal Curve: probability of any continuous random variable taking an exact value.

mean plus minus sd

68%

mean plus minus 2 sigma

95%

mean plus minus 3 sigma

99.7%

Mean of a standard normal distribution.

Standard deviation of a standard normal distribution.

Formula of Z.

Z = x - mean all over sigma

Notation of Standard Normal Distribution.

Z~N(0,1)

Z-table provides values for only the ____ side.

Left

First Rule for Computing Probabilities with Z.

P(Z < z)

Second Rule for Computing Probabilities with Z.

P(Z>z) = 1-P (Z

Third Rule for Computing Probabilities with Z.

P(Z>-z) = P(Z

4th Rule for Computing Probabilities with Z.

P(Z<-z) = P(Z>z) = 1-P(Z

5th Rule for Computing Probabilities with Z.

P(z1 < Z < z2) = P(Z < z2) - P(Z

What is the continuous counterpart of geometric distribution?

Exponential Probability Distribution

Mean and Variance of Exponential Probability Distribution.

1/lambda