Probability: Many Random Variables (Part 2)

1. Probability: Many Random Variables (Part 2) Mike Wasikowski June 12, 2008

2. Contents • Indicator RV’s • Derived RV’s • Order RV’s • Continuous RV Transformations

3. Indicator RV’s • IA = 1 if event A occurs, 0 if not • Consider A1, A2, …, An events, I1, I2, …, In their indicator RV’s, and p1, p2, …, pn the probabilities of events Ai occurring • Then Σj Ijis the total number of events that occur • Mean of a sum of RV’s = sum of the mean of the RV’s (regardless of dependence), so E(Σj Ij) = Σj E(Ij) = Σj pj

4. Indicator RV’s • If all values of pi are equal, then E(Σj Ij) = np • When all events are independent, we calculate variance of number of events that occur as p1(1-p1)+…+pn(1-pn) • If all values of pi are equal and all events are independent, variance is np(1-p) • Thus, we have a binomial distribution

5. Ex: Sequencing EST Libraries • Transcription: DNA → mRNA → amino acids/proteions • EST (expressed sequence tag): a sequence of 100+ base pairs of mRNA • Different genes get expressed at different levels inside a cell • Abundance class L: where a cell contains L copies of an mRNA “species” • Generate an EST DB by sampling with replacement from the mRNA pool, see less rare species less often • How does the number of samples affect the proportion of rare species we will see?

6. Ex: Sequencing EST Libraries • Using indicator RV’s makes this problem easy to solve • Let Ia = 1 if a is in the S samples, 0 if not • Number of species in abundance class L = Σa Ia • We know each Ia has the same mean, so E(Σa Ia) = nLpL

7. Ex: Sequencing EST Libraries • Let pL = 1-rL, where rL is the probability this species is not in the database • rL = (1-L/N)S • Thus, we getE(Σa Ia) = nL(1- (1-L/N)S)

8. Derived RV’s • Previously saw how we find joint distributions and density functions • These joint pdf’s can be used to define many new RV’s • Sum • Average • Orderings • Because many statistical operations use these RV’s, knowing properties of their distributions is important

9. Sums and Averages • Two most important derived RV’s • Sn = X1+X2+…+Xn • X = Sn/n • Mean of Sn = nμ, variance = nσ2 • Mean of X = μ, variance = σ2/n • These properties generalize to well-behaved functions of RV’s and vectors of RV’s as well • Many important applicationsin probability and statisticsuse sums and averages

10. Central Limit Theorem • If X1, X2, ..., Xn are iid with a finite mean and variance, as n→∞, the standardized RV (X-μ)sqrt(n)/σ converges to an RV ~ N(0,1) Image from Wikipedia: Central Limit Theorem

11. Order Statistics • Involve the ordering of n iid RV’s • Call smallest X(1), next smallest X(2), up to biggest X(n) • Xmin =X(1), Xmax =X(n) • We know that these order statistics are distinct because P(X(i) = X(j)) = 0 for independent continuous RV’s

12. Minimum RV (Xmin) • Let X1, X2, ..., Xn be iid as X • If Xmin≥ x, then for each Xi, Xi≥ x • P(Xmin≥ x) = P(X ≥ x)n, also written as 1-Fmin(x) = (1-FX(x)) • By differentiating, we get the density function • fmin(x) = n fX(x)(1-FX(x))n-1

13. Maximum RV (Xmax) • Let X1, X2, ..., Xn be iid as X • If Xmax≤ x, then for each Xi, Xi≤ x • P(Xmax≤ x) = P(X ≤ x)n, also written as Fmax(x) = (FX(x))n • By differentiating, we get the density function • fmin(x) = n fX(x)(FX(x))n-1

14. Density function of X(i) • Let h be a small value, ignore events of probability o(h) • Consider the event that u < X(i) < u+h • In this event, i-1 RV's are less than u, one is between u and u+h, the remaining exceed u+h • Multinomial event with n trials and 3 outcomes • We have an approximation of P(u < X < u+h) ~ fX(u)h

15. Density function of X(i)

16. Continuous RV Transformations • Consider n continuous RV's, X1, X2, ..., Xn • let V1 = V1(X1, X2, ..., Xn),V2, ..., Vn defined similarly • we then have a mapping from (X1, X2, ..., Xn) to (V1, V2, ..., Vn) • If the mapping is 1-1 and differentiable with a differentiable inverse, we can define the Jacobian matrix • Jacobian transformations are used to find marginal functions of one RV when that would be otherwise difficult • Used in ANOVA as well as BLAST

17. Questions?