SOME MOST POPULAR DISTRIBUTIONS OF RANDOM VARIABLES

Transcription

1 SOME MOST POPULAR DISTRIBUTIONS OF RANDOM VARIABLES

2 ... OF THE DISCRETE TYPE 1.ONE-POINT (single-valued) RV: P(X = x 0 ) = 1 { 0 x x0 F (x) = 1 x > x 0 E{X} = x 0 ; VAR(X) = 0. 2.TWO-POINT (two-valued): P(X = x 1 ) = p, P(X = x 2 ) = q = 1 p let s put: x 1 = 1, x 2 = 0 E{X} = p, V AR(X) = p(1 p), µ 3 = p(1 p)(1 2p)

3 BERNOULLI (BINOMIAL) DISTRIBUTION E = A + Ā; P(A) = p; P(Ā) = q = 1 p RV X = i X i a Bernoulli sequence of trials: { 1 if A X i = 0 if Ā ( Wk n n P(X = k) = k ) p k q n k E{X i } = p 1 + q 0 = p; V AR(X i ) =... = pq E{X} = np σ 2 (X) = npq (the variables X i are independent of each other!)

4 BERNOULLI, cntd. A Bernoulli trial process is a sequence of independent and identically distributed RVs: X 1,..., X n n repetitions of an experiment, under identical conditions, with each experiment producing only two outcomes: success (P = p) or failure (P = q = 1 p).

5 The binomial (Bernoulli) distribution... here come few graphs of binomial (Bernoulli) distribution for different p and n values:

6 The binomial (Bernoulli) distribution... more graphs of binomial (Bernoulli) distribution for different p and n values:

7 The binomial (Bernoulli) distribution... more graphs of binomial (Bernoulli) distribution for different p and n values: (W.A. Rosenkrantz, Introduction to Probability and Statistics, Mc-Graw-Hill, 1997)

8 The binomial (Bernoulli) distribution... more graphs of binomial (Bernoulli) distribution for different p and n values: (W.A. Rosenkrantz, Introduction to Probability and Statistics, Mc-Graw-Hill, 1997)

9 The binomial (Bernoulli) distribution... one more graph of binomial (Bernoulli) distribution for different p and n values: (W.A. Rosenkrantz, Introduction to Probability and Statistics, Mc-Graw-Hill, 1997)

10 MULTINOMIAL DISTRIBUTION we have so E = A 1 + A A N ; P(A k ) = p k ; p 1 p 1...p }{{} 1 p 2 p 2...p 2 p }{{} 3 p 3...p }{{} 3 k 1 k 2 the expected value: k 3 k 1 + k 2 + k k N = n W n k 1,k 2,...,k n = n! N j=1 k j! N p k = 1 k=1... p N p N...p N }{{} k N N j=1 p kj j { 1 the outcome of i-th trial = Aj X ij = 0 otherwise n X j = i=1 X ij E{X j } = ˆx j = n p j

11 MULTINOMIAL DISTRIBUTION, cntd. covariances and variances: c ij = np i (δ ij p j ) this means that any two events {A i,a j } cannot be independent (unless p i 0 or p j 0.) p j probability of the event A j can be associated with the frequency ν j ν: ν = 1 n X ij = 1 n n X j i=1 E{ν} = 1 n E{X j} = p j what about the error? ( ) σ 2 (ν) = σ 2 Xj = 1 n n 2 σ2 (X j ) = 1 n 2 np j(1 p j ) = 1 n p j(1 p j ) σ(ν) 1 n

12 MULTINOMIAL DISTRIBUTION, cntd. this is one of the conclusions which may be drawn from the so called LAW OF BIG NUMBERS: the error acompanying an estimation based on an ensemble of size n is 1/ n.

13 HYPERGEOMETRIC DISTRIBUTION Imagine a bag with R red balls and N R black balls (total N balls). We select at random a sample of size n. If we denote by x the number of red balls we have: max(0, n (N R)) x min(n, R) also ( R N R ) h(x) = x)( n x ( N. n) The expected value and variance are E(X) = n R N V AR(X) = N n N 1 R N ( 1 R ). N When the sample size (n) is less than 5 percent of the population size (N) the Hypergeometric Distribution is quite well approximated by the binomial (Bernoulli) distribution with p = R/N.

14 Example: Acceptance sampling (after W.Rosenkrantz) Computer mice are packed in lots of 100. Ten mice are selected and tested. Any one (or more) is found to be ill-functioning the whole lot is rejected, i.e. the lot may be accepted only if none of the tested mice is defective. There are 6 defective mice in the lot. What is the probability of accepting the whole lot? ( 6 94 ) 0)( P(accepting) = P(X = 0) = 10 ( P 0.51; what if we increase the sample size n = 20? ( 6 94 ) 0)( 20 ) roughly 25 percent... ( ).