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Continuous ProbabilityDistributionsUniform Distribution

Important Terms & Concepts Learned Probability Mass Function (PMF) Cumulative Distribution Function (CDF) Complementary Cumulative Distribution Function (CCDF) Expected valueMeanVarianceStandard deviation Uniform distributionBernoulli distribution/trialBinomial distributionPoisson distributionGeometric distributionNegative binomial distribution2

Which distribution is this?A.B.C.D.E.UniformBinomialGeometricNegative BinomialPoissonGet your i‐clickers3

Which distribution is this?A.B.C.D.E.UniformBinomialGeometricNegative BinomialPoissonGet your i‐clickers4

Which distribution is this?A.B.C.D.E.UniformBinomialGeometricNegative BinomialPoissonGet your i‐clickers5

Which distribution is this?A.B.C.D.E.UniformBinomialGeometricNegative BinomialPoissonGet your i‐clickers6

Which distribution is this?A.B.C.D.E.UniformBinomialGeometricNegative BinomialPoissonGet your i‐clickers7

Which distribution is this?A.B.C.D.E.UniformBinomialGeometricNegative BinomialPoissonGet your i‐clickers8

Continuous & Discrete RandomVariables A discrete random variable is usually integernumber– N – the number of proteins in a cell– D‐ number of nucleotides different between twosequences A continuous random variable is a real number– C N/V – the concentration of proteins in a cell ofvolume V– Percentage D/L*100% of different nucleotides inprotein sequences of different lengths L(depending on set of L’s may be discrete but dense)Sec 2‐8 Random Variables10

Probability Mass Function (PMF) X – discrete randomvariable Probability MassFunction: f(x) P(X x)– the probability thatX is exactly equal to xProbability Mass Function forthe # of mismatches in 4‐mersP(X 0) P(X 1) P(X 2) P(X 3) P(X 4) 0.65610.29160.04860.00360.0001Σx P(X x) 1.000011

Probability Density Function (PDF)Density functions, in contrast to mass functions,distribute probability continuously along an intervalFigure 4‐2 Probability is determined from the area under f(x) from a to b.Sec 4‐2 Probability Distributions &Probability Density Functions12

Probability Density FunctionFor a continuous random variable X ,a probability density function is a function such that(1)f x 0means that the function is always non-negative. (2) f ( x)dx 1 b(3)P a X b f x dx area under f x dx from a to baSec 4‐2 Probability Distributions &Probability Density Functions13

Histogram approximates PDFA histogram is graphical display of data showing a series of adjacentrectangles. Each rectangle has a base which represents aninterval of data values. The height of the rectangle creates anarea which represents the probability of X to be within the base.When base length is narrow, the histogram approximates f(x) (PDF):height of each rectangle its area/length of its base.Figure 4‐3 Histogram approximates a probability density function.Sec 4‐2 Probability Distributions &Probability Density Functions16

Cumulative Distribution Functions (CDF & CCDF)The cumulative distribution function (CDF)of a continuous random variable X is,F x P X x x f u dufor x (4-3) One can also use the inverse cumulative distribution functionor complementary cumulative distribution function (CCDF) F x P X x f u du for x xDefinition of CDF for a continous variable is the sameas for a discrete variableSec 4‐3 Cumulative Distribution Functions17

Density vs. Cumulative Functions The probability density function (PDF) is thederivative of the cumulative distributionfunction (CDF).dF x dF x f x dxdxas long as the derivative exists.Sec 4‐3 Cumulative Distribution Functions18

Mean & VarianceSuppose X is a continuous random variable withprobability density function f x . The mean orexpected value of X , denoted as or E X , is E X xf x dx(4-4) The variance of X , denoted as V X or 2 , is 2 V X x f x dx 2 x 2 f x dx 2 The standard deviation of X is 2 .Sec 4‐4 Mean & Variance of a ContinuousRandom Variable19

Gallery of UsefulContinuous Probability Distributions

Continuous Uniform Distribution This is the simplest continuous distributionand analogous to its discrete counterpart. A continuous random variable X withprobability density functionf(x) 1 / (b‐a) for a x b(4‐6)Compare todiscretef(x) 1/(b‐a 1)Figure 4‐8 Continuous uniform PDFSec 4‐5 Continuous Uniform Distribution21

Comparison between Discrete &Continuous Uniform DistributionsDiscrete: PMF: f(x) 1/(b‐a 1) Mean and Variance:μ E(x) (b a)/2σ2 V(x) [(b‐a 1)2–1]/12Continuous: PMF: f(x) 1/(b‐a) Mean and Variance:μ E(x) (b a)/2σ2 V(x) (b‐a)2/1222

X is a continuous random variablewith a uniform distributionbetween 0 and 3.What is P(X 1)?A.B.C.D.E.1/41/30InfinityI have no ideaGet your i‐clickers23

X is a continuous random variablewith a uniform distributionbetween 0 and 3.What is P(X 1)?A.B.C.D.E.1/41/30InfinityI have no ideaGet your i‐clickers24

X is a continuous random variablewith a uniform distributionbetween 0 and 3.What is P(X 1)?A.B.C.D.E.1/41/30InfinityI have no ideaGet your i‐clickers25

X is a continuous random variablewith a uniform distributionbetween 0 and 3.What is P(X 1)?A.B.C.D.E.1/41/30InfinityI have no ideaGet your i‐clickers26

Matlab exercise: generate 100,000 random numbers drawnfrom uniform distribution between 3 and 7 plot histogram approximating its PDF calculate mean, standard deviation andvariance

Matlab template: Uniform PDF Stats ?;r2 ? (std(r2));step 0.1;[a,b] hist(r2,0:step:8);pdf e a./sum(a).? (* or /) step;figure; plot(b,pdf e,'ko‐');

Credit: XKCDcomics

Matlab exercise: Uniform PDF Stats 100000;r2 3 (std(r2));step 0.1;[a,b] hist(r2,0:step:8);pdf e a./sum(a)./step;figure; plot(b,pdf e,'ko‐');

Continuous Uniform Distribution This is the simplest continuous distribution and analogous to its discrete counterpart. A continuous random variable Xwith probability density function f(x) 1 / (b‐a) for a x b (4‐6) Sec 4‐5 Continuous Uniform Distribution 21 Figure 4‐8 Continuous uniform PDF

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