Sampling

1
Sampling

• Random Sample A sample of size n from a
  population with N elements is called a random
  sample if every sample of size n has an equal
  chance of being selected
• Sampling with replacement A sample in which
  members are selected sequentially with each
  element replaced prior to the next selection.
• Sampling without replacement A sample
  consisting of n distinct elements chosen from the
  population.

2
Chapter 3

• Discrete Random Variables and Their Distribution

3
Definitions/Notation

• Discrete Random Variable A random variable, Y,
  is discrete it the range of Y is finite or
  discrete.
• Notation Let Y be a random variable defined on
  a sample space S. The set (Yy) will denote s?S
  Y(s) y
• Probability Function (Probability Mass
  Function) Let Y be a discrete random variable.
  The function defined by p(y) P(Yy) for any
  real number y, is called the probability
  functions or probability mass function for Y.

4
Properties of the Probability Function

• For any discrete random variable Y with
  probability function p(y).
• 0??p(y) ?1
• ?p(y) 1 where the summation is over all
  elements in the range of Y

5
Probability function for g(Y)

• Let Y be a discrete random variable with
  probability function pY(y) and let g be a real
  valued function of a real variable. Suppose the
  random variable X is defined by X g(Y). Then,
  for each x in the range of the random variable X
  the probability function of X is given by pX(x)
  ?pY(yi) where the sum is take over all yi such
  that g(yi) x.

6
Expected Values of Functions of Random Variables

• If Y be a discrete random variable with
  probability function pY(y) and if g be a real
  valued function of a real variable then E(X)
  E(g(Y)) ? g(y) pX(y) where the sum is take
  over all yi in the range of Y. (provided the
  series converges absolutely). If the series does
  not converge absolutely, we say that the expected
  value of X does not exist.

7
Mean, Standard Deviation and Variance

• Mean of Y Let Y be a random variable and
  suppose that the E(Y) exists. Then the mean of
  Y, ? , is defined by ? E(Y)
• Variance of Y Let Y be a random variable. If ?
  exists and if E(Y- ?)2 exists then the
  variance of Y, V(Y) or ?2 , is defined by ?2
  E(Y- ?)2
• Standard Deviation of Y The standard deviation
  of Y, ?, is defined by ? ? ?2

8
Moment Generating Function

• Definition The moment generating function (MGF)
  for a random variable Y is defined by, m(t)
  E(etY) provided that this function exists in an
  interval b lt t lt b for some positive number b.
• Theorem Let Y be a random variable and suppose
  that m(t) exists for t lt b (bgt0). Then E(Ym)
  exists for any positive integer m and E(Ym)
  dm(m(t))/dtmt0.

9
Theorems

• Let Y be discrete random variable with
  probability function p(y).
• Let c be a constant, then E(c) c.
• Let c be a constant and let g be a real valued
  function of a real variable, then E(cg(Y))
  cE(g(Y)
• Let c1 and c2 be constants and let g1 and g2 be
  real valued functions, then E(c1g1(Y) c2g2(Y))
  c1E(g1(Y)) c2E(g2(Y))
• If V(Y) exists, then V(Y) EY2 - ? 2

10
The Binomial Experiment

• Definition A binomial experiment possesses the
  following properties
• The experiment consists of n identical trials.
• Each trial results in one of two outcomes
  success, S or failure F.
• For each trial P(success) p P(failure) q
  1-p where 0 lt p lt 1
• The n trials are independent
• The random variable Y of interest counts the
  number of successes in the n trials.

11
Binomial Random Variable

• The random variable Y which counts the number of
  successes in a Binomial Experiment is said to
  have a Binomial Distribution and the probability
  function for Y is given byp(y)
  C(n,y)py(1-p)n-y for y 0, 1, 2, , n (where
  C(n,y) is the number of combinations of n things
  taken y at a time.)
• Furthermore any random variable with the above
  probability function is said to have a Binomial
  distribution.

12
MGF, Mean and Variance for aBinomial Random
Variable

• Theorem Let Y be a binomial random variable
  based on n trials with success probability p.
  Then
• m(t) (pet (1-p))n for all t
• E(Y) np
• E(Y2) np n(n-1)p2
• V(Y) np(1-p)

13
The Geometric Distribution

• Consider a sequence of Binomial experiments with
  P(success) p (0 lt p lt 1) and P(failure) q 1
  - p. Let Y be the number of trials needed to get
  a success. Then Y is said to have a geometric
  probability distribution
• The pmf of X is given by,p(y) qy-1p for y
  1, 2, 3,

14
MGF, Mean and Variance for aGeometric Random
Variable

• Theorem Let Y be a geometric random variable
  with P(success) p. Then
• m(t) pet/(1- qet) for all t lt -ln(q)
• E(Y) 1/p
• E(Y2) (2-p)/p2
• V(Y) (1-p)/p2

15
Hypergeometric Distribution

• Consider an experiment consisting of selecting a
  sample of size n from a population of N objects
  of which r are of type I and N-r are of type II.
  Let Y denote the number of objects selected of
  type I. Then Y is said to have a hypergeometric
  probability distribution.
• The pmf of Y is givenwhere y is an integer 0,
  1, 2, , n,subject to the condition y ? r andn
  - y ? N - r.

16
Mean and Variance for aHypergeometric Random
Variable

• Theorem Let Y be a hypergeometric random
  variable. Then
• E(Y) nr/N
• V(Y) n ? (r/N) ? (N-r)/(N) ? (N-n)/(N-1)

17
Poisson Process

• A Poisson process consists of an experiment where
  a certain event occurs randomly with the
  following properties
• The number of occurrences in an interval t, t
  ?t is approximately proportional to the length
  of the interval, ?t.
• The number of occurrences in the interval t, t
  ?t does not vary with time t.
• The number of occurrences in non- overlapping
  intervals is independent
• The probability of two or more occurrences in a
  short interval t, t ?t is negligible as ?t?0.
  

18
Poisson Distribution

• Let Y(t) denote the number of events occurring in
  a Poisson process in the time interval 0, t and
  let ? be the average number of occurrence in an
  interval of unit time. Then Y has a Poisson
  distribution with pmf p(y) e-?t(?t)y/y! y
  0, 1, 2, ? gt 0
• If ? is the average number of occurrences in a
  fixed interval then p(y) e-?(?)y/y! y 0,
  1, 2, ? gt 0

19
MGF, Mean and Variance for aPoisson Random
Variable

• Let Y be a random variable with Poisson
  distribution p(y) e-?(?)y/y! for y 0, 1, 2,
  , ? gt 0. Then,
• m(t) exp?(et - 1) for all t
• E(Y) ?
• E(Y2) ? ?2
• V(Y) ?

20
Moments of a Random Variable

• Definition The kth moment of a random variable
  Y taken about the origin is defined to be E(Yk)
  and denoted by ??k provided it exists.
• Definition The kth moment of a random variable
  Y taken about the mean, or the kth central moment
  of Y, is defined to be E((Y-?)k) and denoted by
  ?k provided it exists.

21
Moment Generating Function

• Recall The moment generating function (MGF) for
  a random variable Y is m(t) E(etY) provided
  it exists for t lt b where b gt 0.
• Theorem If m(t) exists for Y, then E(Ym) exists
  and E(Ym) dm(m(t))/dtmt0 for m 1, 2, 3, ...
• Theorem (Uniqueness) If the MGF exists for
  some probability distribution, then it is unique.
  i.e. if the MGF for Y and Z exist and are equal
  (for all tltb, b positive), then Y and Z have
  the same distribution.

22
Tchebysheffs Theorem (Chebyshevs Inequality)

• Theorem Let Y be any random variable with mean
  ? and variance ?2. Then, for any positive
  constant k, P(Y- ? lt k?) ? 1 (1/k2)or
  P(Y- ? ? k?) ? (1/k2).

23
Homework

• p.129 98, 99, 100, 104, 106, 109, 119, 120,
  121, 123