Continuous Distributions

Transcription

1 Continuous Distributions : Continuous Random Variables : Uniform Distribution (Continuous) Exponential and Gamma Distributions: Distance between crossovers Prof. Tesler Math 283 Fall 2015 Prof. Tesler Continuous Distributions Math 283 / Fall / 24

2 Continuous distributions Example Pick a real number x between 20 and 30 with all real values in [20, 30] equally likely. Sample space: S = [20, 30] Number of outcomes: S = Probability of each outcome: P(X = x) = 1 = 0 Yet, P(X 21.5) = 15% Prof. Tesler Continuous Distributions Math 283 / Fall / 24

3 Continuous distributions The sample space S is often a subset of R n. We ll do the 1-dimensional case S R. The probability density function (pdf) f X (x) is defined differently than the discrete case: f X (x) is a real-valued function on S with f X (x) 0 for all x S. S f X (x) dx = 1 (vs. x S P X (x) = 1 for discrete) The probability of event A S is P(A) = f X (x) dx (vs. P X (x)). A x A In n dimensions, use n-dimensional integrals instead. Uniform distribution Let a < b be real numbers. The Uniform Distribution on [a, b] is that all numbers in [a, b] are equally likely. { 1 More precisely, f X (x) = b a if a x b; 0 otherwise. Prof. Tesler Continuous Distributions Math 283 / Fall / 24

4 Uniform distribution (real case) The uniform distribution on [20, 30] We could regard the sample space as [20, 30], or as all reals. f X (x) = { 1/10 for 20 x 30; 0 otherwise. f X (x) P(X 21.5) = 20 0 dx = 10 =.15 = 15% dx = 0 + x f X (x) x Prof. Tesler Continuous Distributions Math 283 / Fall / 24 x

5 Cumulative distribution function (cdf) The Cumulative Distribution Function (cdf) of a random variable X is (x) = P(X x) For a continuous random variable, (x) = P(X x) = x f X (t) dt and f X (x) = F (x) X The integral cannot have x as the name of the variable in both of (x) and f X (x) because one is the upper limit of the integral and the other is the integration variable. So we use two variables x, t. We can either write or (x) = P(X x) = (t) = P(X t) = x t f X (t) dt f X (x) dx Prof. Tesler Continuous Distributions Math 283 / Fall / 24

6 CDF of uniform distribution Uniform distribution on [20, 30] For x < 20: (x) = x 0 dt = 0 For 20 x < 30: (x) = 20 0 dt + x 20 1 x 20 10dt = 10 For 30 x: (x) = 20 0 dt dt + x 30 0 dt = 1 Together: 0 if x < 20 (x)= x if 20 x 30 1 if x 30 0 if x < 20 f X (x)=f X (x)= 1 10 if 20 x 30 0 if x 30 Prof. Tesler Continuous Distributions Math 283 / Fall / 24

7 PDF vs. CDF f X (x) Probability density function {.1 if 20 x 30; f X (x)= 0 otherwise. It s discontinuous at x = 20 and 30. PDF is derivative of CDF: f X (x) = (x) x (x) Cumulative distribution function (x) = 0 if x < 20; (x 20)/10 if 20 x 30; 1 if x 30. CDF is integral of PDF: (x) = x x f X (t) dt Prof. Tesler Continuous Distributions Math 283 / Fall / 24

8 PDF vs. CDF: Second example Probability density function Cumulative distribution function density f R (r) F R (r) r { 2r/9 if 0 r < 3; f R (r)= 0 if r 0 or r > 3 It s discontinuous at r = 3. PDF is derivative of CDF: f R (r) = F R (r) r 0 if r < 0; F R (r) = r 2 /9 if 0 r 3; 1 if r 3. CDF is integral of PDF: F R (r) = r f R (t) dt Prof. Tesler Continuous Distributions Math 283 / Fall / 24

9 Probability of an interval Compute P( 1 R 2) from the PDF and also from the CDF Computation from the PDF P( 1 R 2) = = = 0 + f R (r) dr = 0 dr + ( r r= r 9 dr ) f R (r) dr + = = 4 9 f R (r) dr Computation from the CDF P( 1 R 2) = P( 1 < R 2) = F R (2) F R ( 1 ) = = 4 9 Prof. Tesler Continuous Distributions Math 283 / Fall / 24

10 Continuous vs. discrete random variables Cumulative distribution function Cumulative distribution function 1 1 F R (r) 0.5 (x) r In a continuous distribution: 0! x The probability of an individual point is 0: P(R = r) = 0. So, P(R r) = P(R < r), i.e., F R (r) = F R (r ). The CDF is continuous. (In a discrete distribution, the CDF is discontinuous due to jumps at the points with nonzero probability.) P(a < R < b)= P(a R < b) = P(a < R b) = P(a R b) = F R (b) F R (a) Prof. Tesler Continuous Distributions Math 283 / Fall / 24

11 Cumulative distribution function (cdf) The Cumulative Distribution Function (cdf) of a random variable X is (x) = P(X x) Continuous case (x) = x f (t) dt X Weakly increasing. Varies smoothly from 0 to 1 as x varies from to. To get the pdf from the cdf, use f X (x) = (x). Discrete case (x) = t x P X (t) Weakly increasing. Stair-steps from 0 to 1 as x goes from to. The cdf jumps where P X (x) 0 and is constant in-between. To get the pdf from the cdf, use P X (x) = (x) (x ) (which is positive at the jumps, 0 otherwise). Prof. Tesler Continuous Distributions Math 283 / Fall / 24

12 CDF, percentiles, and median The kth percentile of a distribution X is the point x where k% of the probability is up to that point: (x) = P(X x) = k% = k/100 Example: F R (r) = P(R r) = r 2 /9 (for 0 r 3) r 2 /9 = (k/100) r = 9(k/100) 75th percentile: r = 9(.75) 2.60 Median (50th percentile): r = 9(.50) th and 100th percentiles: r = 0 and r = 3 if we restrict to the range 0 r 3. But they are not uniquely defined, since F R (r) = 0 for all r 0 and F R (r) = 1 for all r 3. Prof. Tesler Continuous Distributions Math 283 / Fall / 24

13 Expected value and variance (continuous r.v.) Replace sums by integrals. It s the same definitions in terms of E( ) : µ = E(X) = E(g(X)) = x f X (x) dx g(x) f X (x) dx σ 2 = Var(X) = E((X µ) 2 ) = E(X 2 ) (E(X)) 2 µ and σ for the uniform distribution on [a, b] (with a < b) b 1 µ = E(X) = x a b a dx = x2 b /2 b a = (b2 a 2 )/2 = b + a x=a b a 2 b E(X 2 ) = x 2 1 a b a dx = x3 b /3 b a = (b3 a 3 )/3 = b2 + ab + a 2 x=a b a 3 σ 2 = Var(X) = E(X 2 ) (E(X)) 2 = b2 + ab + a 2 ( ) b + a 2 (b a)2 = σ = SD(X) = (b a)/ 12 Prof. Tesler Continuous Distributions Math 283 / Fall / 24

14 Exponential distribution How far is it from the start of a chromosome to the first crossover? How far is it from one crossover to the next? Let D be the random variable giving either of those. It is a real number > 0, with the exponential { distribution λ e λ d if d 0; f D (d) = 0 if d < 0. where crossovers happen at a rate λ = 1 M 1 = 0.01 cm 1. General case Crossovers Mean E(D) = 1/λ = 100 cm = 1 M Variance Var(D) = 1/λ 2 = cm 2 = 1 M 2 Standard Dev. SD(D) = 1/λ = 100 cm = 1 M Prof. Tesler Continuous Distributions Math 283 / Fall / 24

15 Exponential distribution Exponential distribution µ µ±! Exponential: "=0.01 pdf d Prof. Tesler Continuous Distributions Math 283 / Fall / 24

16 Exponential distribution In general, if events occur on the real number line x 0 in such a way that the expected number of events in all intervals [x, x + d] is λ d (for x > 0), then the exponential distribution with parameter λ models the time/distance/etc. until the first event. It also models the time/distance/etc. between consecutive events. Chromosomes are finite; to make this model work, treat there is no next crossover as though there is one but it happens somewhere past the end of the chromosome. Prof. Tesler Continuous Distributions Math 283 / Fall / 24

17 Proof of pdf formula Let d > 0 be any real number. Let N(d) be the # of crossovers that occur in the interval [0, d]. 0 d D>d D<d D<d N(d)=0 N(d)=1 N(d)=2 If N(d) = 0 then there are no crossovers in [0, d], so D > d. If D > d then the first crossover is after d so N(d) = 0. Thus, D > d is equivalent to N(d) = 0. P(D > d) = P(N(d) = 0) = e λ d (λ d) 0 /0! = e λ d since N(d) has a Poisson distribution with parameter λ d. The cdf of D is F D (d) = P(D d) = 1 P(D > d) = { 1 e λ d if d 0; 0 if d < 0. Differentiating the cdf gives pdf f D (d) = F D (d) = λ e λ d (if d 0). Prof. Tesler Continuous Distributions Math 283 / Fall / 24

18 Discrete and Continuous Analogs Success Discrete Coin flip at a position is heads Continuous Point where crossover occurs Rate Probability p per flip λ (crossovers per Morgan) # successes Binomial distribution: Poisson distribution: Wait until 1st success Wait until rth success # heads out of n flips # crossovers in distance d Geometric distribution Negative binomial distribution Exponential distribution Gamma distribution Prof. Tesler Continuous Distributions Math 283 / Fall / 24

19 Gamma distribution How far is it from the start of a chromosome until the rth crossover, for some choice of r = 1, 2, 3,...? Let D r be a random variable giving this distance. It has the gamma distribution { with pdf λ r (r 1)! f Dr (d) = dr 1 e λ d if d 0; 0 if d < 0. Mean E(D r ) = r/λ Variance Standard deviation Var(D r ) = r/λ 2 SD(D r ) = r/λ The gamma distribution for r = 1 is the same as the exponential distribution. The sum of r i.i.d. exponential variables, D r = X 1 + X X r, each with rate λ, gives the gamma distribution. Prof. Tesler Continuous Distributions Math 283 / Fall / 24

20 Gamma distribution 8/9 *+(!!),-..-+/ )&' µ ) µ±!,-..-:+3;)<+";!&!( "&' " (&' (!&'!! "!! #!! $!! %!! / Prof. Tesler Continuous Distributions Math 283 / Fall / 24

21 Proof of Gamma distribution pdf for r = 3 Let d > 0 be any real number. D 3 > d is the event that the third crossover does not happen until sometime after position d. 0 d D 3>d N(d)=0 D 3>d N(d)=1 D 3>d N(d)=2 D 3<d N(d)=3 D 3<d N(d)=4 When D 3 > d, the number N(d) of crossovers in the chromosome interval [0, d] is less than 3, so it s 0, 1, or 2. D 3 > d is equivalent to N(d) < 3. D 3 d is equivalent to N(d) 3. Prof. Tesler Continuous Distributions Math 283 / Fall / 24

22 Proof of Gamma distribution pdf for r = 3 Let d > 0 be any real number. D 3 > d is the event that the third crossover does not happen until sometime after position d. When D 3 > d, the number N(d) of crossovers in the chromosome interval [0, d] is less than 3, so it s 0, 1, or 2: P(D 3 > d) = P(N(d)=0) ( + P(N(d)=1) + ) P(N(d)=2) = e λ d (λ d) 0 (λ d)1 (λ d)2 0! + 1! + 2! The cdf of D 3 is P(D 3 d) = 1 P(D 3 > d). Differentiating the cdf and simplifying gives the pdf { λ 3 d 2 e λd /2! if d 0; f D3 (d) = 0 if d < 0. Prof. Tesler Continuous Distributions Math 283 / Fall / 24

23 The Gamma function and factorials The Gamma function is a generalization of factorials: Γ(z) = for real z > 0. 0 t z 1 e t dt Γ(z) = (z 1)! for z = 1, 2, 3,... Γ(z) extends to all complex 0 numbers except integers Γ(z) = (z 1)! for z = 1, 2, 3,.... Γ(1) = 0 Gamma(z) t 0 e t dt = e t 0 = = 1 Γ(z) = (z 1)Γ(z 1) can be shown using integration by parts: differentiate t z 1 and integrate up e t dt. When z is a positive integer, iterate this to Γ(z) = (z 1)(z 2) (2)(1)Γ(1) = (z 1)! Γ(1) = (z 1)! Prof. Tesler Continuous Distributions Math 283 / Fall / 24 5 z

24 Variations of the distributions The Gamma distribution is defined for real r > 0 rather than just positive integers: f Dr (d) = { λ r Γ (r) dr 1 e λ d if d 0; 0 if d < 0. (The denominator (r 1)! was replaced by Γ(r).) For Poisson, Exponential, and Gamma distributions, instead of the rate parameter λ, some people use the shape parameter θ = 1/λ: For crossovers, θ = 1 M = 100 cm. The Poisson parameter for distance d is µ = λ d = d/θ. Prof. Tesler Continuous Distributions Math 283 / Fall / 24