Discrete Probability Distributions

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Tutorial 1

• Discrete Probability Distributions

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Bernoulli Distribution

• The outcome of some kind of trials can be either
  a success or a failure. (e.g. tossing a coin)
• Denote X 1 if the outcome is a success and X
  0 if the outcome is a failure.
• If P(X 1) p, then P(X 0) 1 - p.
• We said X is a Bernoulli random variable.

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Bernoulli Distribution

• Mean of X EX (1-p)0 p1
• p
• Variance of X Var(X) (1-p)(0-EX)2
• p(1-EX)2
• (1-p)p

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Binomial Distribution

• If we perform Bernoulli trials n times, then the
  no. of successes X is binomial distributed.
• It is usually denoted by
• where n total no. of trials
• p probability of success

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Binomial Distribution

• Each trial in binomial distribution is
  independent. (i.e. outcome of each trial will not
  affect others)
• n is a fixed no.
• When n 1, the binomial distribution is reduced
  to a Bernoulli distribution.
• EX np, Var(X) np(1-p)

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Binomial Distribution

• E.g. What is the probability that we get 4 heads
  after tossing a coin 10 times?

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Negative Binomial Distribution

• Now, suppose the no. of trials is not fixed.
• We perform Bernoulli trials repeatedly until a
  given no. of successes r are observed. Then we
  stop the trials.
• We said the no. of trials required X is a
  negative binomial random variable

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Negative Binomial Distribution

• There are (r-1) successes in the first (n-1)
  trials.
• The n-th (last) trial must be a success.
• EX r/p, Var(X) r(1-p)/p2

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Negative Binomial Distribution

• E.g. A man practices shooting. If he hits the
  target with probability 0.7, what is the
  probability that he is required to shoot 10 times
  in order to hit the targets 7 times?

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Geometric Distribution

• It is a particular case of -ve binomial distr.
• Suppose we perform Bernoulli trials until a
  success occurs (i.e. r1).
• If we let Y equal the no. of trials required
• Then Y is a geometric random variable.

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Geometric Distribution

• Geometric distribution is memoryless
• Why?

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Geometric Distribution

• How many trials should be made on average in
  order to get a head when tossing a coin?