Continuous Probability Distributions. Uniform Distribution

Transcription

1 Continuous Probability Distributions Uniform Distribution

2 Important Terms & Concepts Learned Probability Mass Function (PMF) Cumulative Distribution Function (CDF) Complementary Cumulative Distribution Function (CCDF) Expected value Mean Variance Standard deviation Uniform distribution Bernoulli distribution/trial Binomial distribution Poisson distribution Geometric distribution Negative binomial distribution 2

3 Which distribution is this? A. Uniform B. Binomial C. Geometric D. Negative Binomial E. Poisson Get your i clickers 3

4 Which distribution is this? A. Uniform B. Binomial C. Geometric D. Negative Binomial E. Poisson Get your i clickers 4

5 Which distribution is this? A. Uniform B. Binomial C. Geometric D. Negative Binomial E. Poisson Get your i clickers 5

6 Which distribution is this? A. Uniform B. Binomial C. Geometric D. Negative Binomial E. Poisson Get your i clickers 6

7 Which distribution is this? A. Uniform B. Binomial C. Geometric D. Negative Binomial E. Poisson Get your i clickers 7

8 Which distribution is this? A. Uniform B. Binomial C. Geometric D. Negative Binomial E. Poisson Get your i clickers 8

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10 Continuous & Discrete Random Variables A discrete random variable is usually integer number N the number of proteins in a cell D number of nucleotides different between two sequences A continuous random variable is a real number C=N/V the concentration of proteins in a cell of volume V Percentage D/L*100% of different nucleotides in protein sequences of different lengths L (depending on set of L s may be discrete but dense) Sec 2 8 Random Variables 10

11 Probability Mass Function (PMF) X discrete random variable Probability Mass Function: f(x)=p(x=x) the probability that X is exactly equal to x Probability Mass Function for the # of mismatches in 4 mers P(X =0) = P(X =1) = P(X =2) = P(X =3) = P(X =4) = Σ x P(X=x)=

12 Probability Density Function (PDF) Density functions, in contrast to mass functions, distribute probability continuously along an interval Figure 4 2 Probability is determined from the area under f(x) from a to b. Sec 4 2 Probability Distributions & Probability Density Functions 12

13 Probability Density Function For a continuous random variable X, a probability density function is a function such that x (1) f 0 means that the function is always non-negative. (2) (3) f( x) dx1 P a X b f x dx b a area under f x dx from a to b Sec 4 2 Probability Distributions & Probability Density Functions 13

14 Normalized histogram approximates PDF A histogram is graphical display of data showing a series of adjacent rectangles. Each rectangle has a base which represents an interval of data values. The height of the rectangle is a number of events in the sample within the base. When base length is narrow, the histogram could be normalized to approximate PDF (f(x)): height of each rectangle = =(# of events within base)/(total # of events)/width of its base. Normalized histogram approximates a probability density function.

15 Cumulative Distribution Functions (CDF & CCDF) The cumulative distribution function (CDF) of a continuous random variable X is, x Fx PX x f udu for x (4-3) One can also use the inverse cumulative distribution function or complementary cumulative distribution function (CCDF) F x P X x f u du for x x Definition of CDF for a continous variable is the same as for a discrete variale b Sec 4 3 Cumulative Distribution Functions 15

16 Density vs. Cumulative Functions The probability density function (PDF) is the derivative of the cumulative distribution function (CDF). df x df x f x =- dx dx as long as the derivative exists. Sec 4 3 Cumulative Distribution Functions 16

17 Mean & Variance Suppose X is a continuous random variable with x probability density function f. The mean or expected value E X xf x dx of X, denoted as or E X, is 2 The variance of X, denoted as V X or, is 2 (4-4) V X x f x dx x f x dx The standard deviation of X is 2. Sec 4 4 Mean & Variance of a Continuous Random Variable 17

18 Gallery of Useful Continuous Probability Distributions

19 Continuous Uniform Distribution This is the simplest continuous distribution and analogous to its discrete counterpart. A continuous random variable X with probability density function f(x) = 1 / (b a) for a x b (4 6) Compare to discrete f(x) = 1/(b a+1) Figure 4 8 Continuous uniform PDF Sec 4 5 Continuous Uniform Distribution 19

20 Comparison between Discrete & Continuous Uniform Distributions Discrete: Continuous: PMF: f(x) = 1/(b a+1) Mean and Variance: μ= E(x) = (b+a)/2 σ 2 = V(x) = [(b a+1) 2 1]/12 PMF: f(x) = 1/(b a) Mean and Variance: μ= E(x) = (b+a)/2 σ 2 = V(x) = (b a) 2 /12 20

21 X is a continuous random variable with a uniform distribution between 0 and 3. What is P(X=1)? A. 1/4 B. 1/3 C. 0 D. Infinity E. I have no idea Get your i clickers 21

22 X is a continuous random variable with a uniform distribution between 0 and 3. What is P(X=1)? A. 1/4 B. 1/3 C. 0 D. Infinity E. I have no idea Get your i clickers 22

23 X is a continuous random variable with a uniform distribution between 0 and 3. What is P(X<1)? A. 1/4 B. 1/3 C. 0 D. Infinity E. I have no idea Get your i clickers 23

24 X is a continuous random variable with a uniform distribution between 0 and 3. What is P(X<1)? A. 1/4 B. 1/3 C. 0 D. Infinity E. I have no idea Get your i clickers 24

25 Credit: XKCD comics

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28 Poisson (constant rate) processes Let s assume that proteins are produced by all ribosomes in the cell at a rate λ per second. The expected number of proteins produced in x seconds is λx. The actual number of proteins N x is a discrete random variable following a Poisson distribution with mean λx: P N (N x =n)=exp( λx)(λx) n /n! E(N x )= λx Why Discrete Poisson Distribution? Divide time into many tiny intervals of length x 0 <<1/ λ The probability of success (protein production) per internal is small: p= λ x 0 <<1, The number of intervals is large: L= x/x 0 >>1 Mean is constant: E(N x )=p L= λ x 0 x/ x 0 =λ x P(N x =n)=l!/n!(l n)! p n (1 p) L n In the limit p0 we have Binomial distribution Poisson

29 Exponential Distribution Definition Exponential random variable X describes interval between two successes of a constant rate (Poisson) random process with success rate λ per unit interval. The probability density function of X is: f(x) = λe λx for 0 x < (4 14) Closely related to the discrete geometric distribution f(x) = p(1 p) x 1 (3 9) Sec 4 8 Exponential Distribution 29

30 What is the interval X between two successes of a constant rate process? X is a continuous random variable CDF: P X (X>x) = P N (N X =0)=exp( λx). Remember: P N (N X =n)=exp( λx) (λx) n /n! PDF: f X (x)= dcdf X (x)/dx = λexp( λx) We started with a discrete Poisson distribution where time x was a parameter We ended up with a continuous exponential distribution

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33 Exponential Mean & Variance If the random variable X has an exponential distribution with parameter, E X and V X (4-15) 2 Note that, for the: Poisson distribution: mean= variance Exponential distribution: mean = standard deviation = variance 0.5 Sec 4 8 Exponential Distribution 33

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35 Biochemical Reaction Time The time x (in minutes) until an enzyme successfully catalyzes a biochemical reaction is approximated by this CDF: F x 1 e for 0 x x/1.4 What is the PDF? df x d x/1.4 x/1.4 f x e e [1 ] /1.4 for 0 x dx dx What proportion of reactions is complete within 0.5 minutes? 0.5/1.4 P X 0.5 F 0.5 1e Sec 4 3 Cumulative Distribution Functions 35

36 The reaction product is overdue : no product has been generated in the past 3 minutes. What is the probability that a product will appear in the next 0.5 minutes? x/1.4 1 e F x F(0.5) 0.3 F(3.5) 0.92 A B. 0.3 C D E. I have no idea Get your i clickers 36

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39 Exponential Distribution in Reliability The reliability of electronic components is often modeled by the exponential distribution. A chip might have mean time to failure of 40,000 operating hours. The memoryless property implies that the component does not wear out the probability of failure in the next hour is constant, regardless of the component age. The reliability of mechanical components do have a memory the probability of failure in the next hour increases as the component ages. Sec 4 8 Exponential Distribution 39

40 Erlang Distribution The Erlang distribution is a generalization of the exponential distribution. The exponential distribution models the time interval to the 1 st event, while the Erlang distribution models the time interval to the r th event, i.e., a sum of r exponentially distributed variables. The exponential, as well as Erlang distributions, is based on the constant rate Poisson process. 40

41 Erlang Distribution Generalizing from the constant rate Poisson Exponential : x r1 x e x PX x k 0 k! 1 Fx Now differentiating F x we find that all terms in the sum except the last one cancel each other: f r r1 x e r 1 x! k for x 0 and r 1,2,3,... Sec 4 9 Erlang & Gamma Distributions 41

42 Gamma Distribution The random variable X with a probability density function: r r1 x x e f x, for x0 (4-18) r has a gamma random distribution with parameters λ > 0 and r > 0. If r is an positive integer, then X has an Erlang distribution. Sec 4 9 Erlang & Gamma Distributions 42

43 Gamma Function The gamma function is the generalization of the factorial function for r > 0, not just non negative integers. r r1 x x e dx r 0, for 0 (4-17) Properties of the gamma function 1 1 r r r r r recursive property 1! factorial function interesting fact Sec 4 9 Erlang & Gamma Distributions 43

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46 Mean & Variance of the Erlang and Gamma If X is an Erlang (or more generally Gamma) random variable with parameters λ and r, μ= E(X) = r / λ and σ 2 = V(X) = r / λ 2 (4 19) Generalization of exponential results: μ= E(X) = 1 / λ and σ 2 = V(X) = 1 / λ 2 or Negative binomial results: μ= E(X) = r / p and σ 2 = V(X) = r(1 p) / p 2 Sec 4 9 Erlang & Gamma Distributions 46

47 Matlab exercise: Generate a sample of 100,000 variables with exponential distribution with λ =0.1 Calculate mean and standard deviation and compare them to 1/ λ Plot semilog y plots of PDF and CCDF. Hint: read the help page for random( Exponential ): it uses something else instead of λ. What?