continuous random variables

Transcription

1 continuous random variables

2 continuous random variables Discrete random variable: takes values in a finite or countable set, e.g. X {1,2,..., 6} with equal probability X is positive integer i with probability 2 -i Continuous random variable: takes values in an uncountable set, e.g. X is the weight of a random person (a real number) X is a randomly selected point inside a unit square X is the waiting time until the next packet arrives at the server 2

3 f(x) : the probability density function (or simply density ) pdf and cdf f(x) a F(a) = f(x) dx a b P(X a) = F(x): the cumulative distribution function (or simply distribution ) P(a < X b) = F(b) - F(a) Need f(x) 0, & f(x) dx (= F(+ )) = 1 - A key relationship: f(x) = d dx + a F(x), since F(a) = f(x) dx, 3

4 Densities are not probabilities densities P(x = a) = P(a X a) = F(a)-F(a) = 0 I.e., the probability that a continuous random variable falls at a specified point is zero P(a - ε/2 X a + ε/2) = f(x) F(a + ε/2) - F(a - ε/2) ε f(a) a-ε/2 a a+ε/2 I.e., The probability that it falls near that point is proportional to the density; in a large random sample, expect more samples where density is higher (hence the name density ). 4

5 sums and integrals; expectation Much of what we did with discrete r.v.s carries over almost unchanged, with Σx... replaced by... dx E.g. For discrete r.v. X, E[X] = Σx xp(x) For continuous r.v. X, Why? (a) We define it that way (b) The probability that X falls near x, say within x±dx/2, is f(x)dx, so the average X should be Σ xf(x)dx (summed over grid points spaced dx apart on the real line) and the limit of that as dx 0 is xf(x)dx 5

6 Let 1 example f(x) F(x)

7 properties of expectation Linearity E[aX+b] = ae[x]+b E[X+Y] = E[X]+E[Y] still true, just as for discrete Functions of a random variable E[g(X)] = g(x)f(x)dx just as for discrete, but w/integral 7

8 variance Definition is same as in the discrete case Var[X] = E[(X-μ) 2 ] where μ = E[X] Identity still holds: Var[X] = E[X 2 ] - (E[X]) 2 proof same 8

9 Let 1 example f(x) F(x)

10 continuous random variables: summary Continuous random variable X has density f(x), and

11 uniform random variable X ~ Uni(α,β) is uniform in [α,β] The Uniform Density Function Uni(0.5,1.0) f(x) α x β

12 uniform random variable X ~ Uni(α,β) is uniform in [α,β] f(x) The Uniform Density Function Uni(0.5,1.0) x if α a b β:

13 X ~ Exp(λ) exponential random variable The Exponential Density Function f(x) ! = 2! = x

14 exponential random variable X ~ Exp(λ) Memorylessness: = 1-F(t)

15 Radioactive decay: How long until the next alpha particle? Examples Customers: how long until the next customer/packet arrives at the checkout stand/server? Buses: How long until the next #71 bus arrives on the Ave? Yes, they have a schedule, but given the vagaries of traffic, riders with-bikes-and-babycarriages, etc., can they stick to it? Relation to the Poisson: Poisson: how many events in a fixed time; Exponential: how long until the next event Relation to geometric: Geometric is discrete analog: How long to a Head, 1 flip per sec, prob p vs How long to a Head, 2 flips per sec, prob p/2,... Limit is exponential with parameter p 15

16 normal random variable X is a normal (aka Gaussian) random variable X ~ N(μ, σ 2 ) The Standard Normal Density Function f(x) µ = 0! = x

17 changing μ, σ µ = 0! = µ = 0! = µ = 4! = µ = 4! = density at μ is.399/σ 17

18 X is a normal random variable X ~ N(μ,σ 2 ) Y = ax + b E[Y] = E[aX+b] = aμ + b Var[Y] = Var[aX+b] = a 2 σ 2 Y ~ N(aμ + b, a 2 σ 2 ) Important special case: Z = (X-μ)/σ ~ N(0,1) normal random variable Z ~ N(0,1) standard (or unit) normal Use Φ(z) to denote CDF, i.e. no closed form

19 able 1: Table of the Standard Normal Cumulative Distribution Func Table of the Standard Normal Cumulative Distribution Function Φ(Z) -0.0z Φ(.46) The Standard Normal Density Function µ = ! = x E.g., see B&T p155, p f(x)

20 The Standard Normal Density Function f(x) µ = 0! = If Z ~N(μ,σ) what is P( μ-σ < Z < μ+σ )? P( μ - σ < Z < μ + σ ) = Φ(1) - Φ(-1) 68% P( μ - 2σ < Z < μ + 2σ ) = Φ(2) - Φ(-2) 95% P( μ - 3σ < Z < μ + 3σ ) = Φ(3) - Φ(-3) 99% x 20

21 normal approximation to binomial X ~ Bin(n,p) E[X] = np Var[X] = np(1-p) Poisson approx: good for n large, p small (np constant) Normal approx: For large n, (p stays fixed): X Y ~ N(E[X], Var[X]) = N(np,np(1-p)) Normal approximation good when np(1-p) 10 DeMoivre-Laplace Theorem: Let S n = number of successes in n trials (with prob. p). Then, as n :

22 normal approximation to binomial P(X=k) Normal(np, np(1-p)) Binomial(n,p) Poisson(np) n = 100 p = k 22

23 normal approximation to binomial Fair coin flipped 40 times. Probability of 20 heads? Exact answer: Normal approximation: 23

24 the central limit theorem (CLT) Consider i.i.d. (independent, identically distributed) random vars X 1, X 2, X 3, X i has μ = E[X i ] and σ 2 = Var[X i ] As n, Restated: As n,

25 How tall are you? Why? Credit: Annie Leibovitz, 1987? Willie Shoemaker & Wilt Chamberlain 25

26 in the real world Human height is approximately normal. Why might that be true? Frequency R.A. Fisher (1918) noted it would follow from CLT if height Male Height in Inches were the sum of many independent random effects, e.g. many genetic factors (plus some environmental ones like diet). I.e., suggested part of mechanism by looking at shape of the curve. (WAY before anyone really knew what genes were...) 26

27 Meta-analysis of Dense Genecentric Association Studies Reveals Common and Uncommon Variants Associated with Height, Lanktree, et al. The American Journal of Human Genetics 88, 6 18, January 7, 2011 Sixty-Four (and hundreds more probably exist) 27

28 in the real world 28

29 in the real world 29

30 in the real world 30

31 pdf and cdf continuous r.v. s: summary d a f(x) = F(x) F(a) = f(x) dx dx sums become integrals, e.g. E[X] = Σx xp(x) most familiar properties still hold, e.g. E[aX+bY+c] = ae[x]+be[y]+c Var[X] = E[X 2 ] - (E[X]) 2 31

32 continuous r.v. s: summary Three important examples X ~ Uni(α,β) uniform in [α,β] E[X] = (α+β)/2 Var[X] = (α-β) 2 /12 f(x) x X ~ Exp(λ) exponential f(x) ! = 2! = x X ~ N(μ, σ 2 ) normal (aka Gaussian) f(x) µ = 0! = x 32