Stat 13 Lecture 19 discrete random variables, binomial - PowerPoint PPT Presentation

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Stat 13 Lecture 19 discrete random variables, binomial

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A random variable is discrete if it takes values that have gaps : most often, integers ... Probability function gives P (X=x) for every value that X may take. X ... – PowerPoint PPT presentation

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Title: Stat 13 Lecture 19 discrete random variables, binomial


1
Stat 13 Lecture 19 discrete random variables,
binomial
  • A random variable is discrete if it takes values
    that have gaps most often, integers
  • Probability function gives P (Xx) for every
    value that X may take
  • X number of heads in 3 tosses
  • A couple decides to have children till at least
    one of each sex or a maximum of 3 Xnumber of
    girls. (tree)

2
Expected value and standard deviation
  • E(X) sum of P(Xx) x (weighted average using
    probability as weight)
  • Var (X) sum of P(Xx) (x- E(X))2

3
Binomial probability
  • Coin tossing multiple choices formula of
    binomial combination number
  • P(Xx) ( ) px (1-p)(n-x)
  • Sampling with replacement
  • Sampling without replacement infinite population
  • Sampling without replacement, finite population
    (opinion ) survey sampling

n x
4
Conditions for binomial to hold
  • Model the number of successful trials out of n
    trials
  • Must know n
  • Must know (or be able to estimate) p (prob of
    success in each trial)
  • Must satisfy independence assumption in different
    trials
  • p should be the same in each trial

5
Sampling without replacement
  • Suppose in a population of N individuals, a
    random sample of n individuals are selected.
    Their opinions on a proposal are recorded.
    Suppose in the population the proportion of
    individuals saying yes is p. Then X, the number
    of individuals in the sample saying yes follows a
    hypergeomtric distribution
  • P(Xx) Np choose xN(1-p) choose (n-x)/ N
    choose n, which is approximately equal to
    binomial when N is large and the sampling
    fraction n/N is small.

6
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