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Tutorial 1Tutorial 1

Discrete Probability DistributionsDiscrete Probability Distributions

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Bernoulli DistributionBernoulli Distribution

The outcome of some kind of trials can be The outcome of some kind of trials can be either a success or a failure. (e.g. tossing a either a success or a failure. (e.g. tossing a coin)coin)

Denote X = 1 if the outcome is a success and Denote X = 1 if the outcome is a success and X = 0 if the outcome is a failure.X = 0 if the outcome is a failure.

If P(X = 1) = p, then P(X = 0) = 1 - p.If P(X = 1) = p, then P(X = 0) = 1 - p. We said X is a Bernoulli random variable.We said X is a Bernoulli random variable.

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Bernoulli DistributionBernoulli Distribution

Mean of X = E[X] = (1-p)*0 + p*1Mean of X = E[X] = (1-p)*0 + p*1

= p= p

Variance of X = Var(X) = (1-p)(0-E[X])Variance of X = Var(X) = (1-p)(0-E[X])22

+ p(1-E[X])+ p(1-E[X])22

= (1-p)p= (1-p)p

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Binomial DistributionBinomial Distribution

If we perform Bernoulli trials If we perform Bernoulli trials nn times, then times, then the no. of successes the no. of successes XX is binomial distributed. is binomial distributed.

It is usually denoted by:It is usually denoted by:

where where nn = total no. of trials = total no. of trials

pp = probability of success = probability of success

b(x, n, p) = P(X = x) =n

x

p p x nx n x1 0,

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Binomial DistributionBinomial Distribution

Each trial in binomial distribution is Each trial in binomial distribution is independentindependent. (i.e. outcome of each trial will . (i.e. outcome of each trial will not affect others)not affect others)

nn is a fixed no. is a fixed no. When When nn = 1, the binomial distribution is = 1, the binomial distribution is

reduced to a Bernoulli distribution.reduced to a Bernoulli distribution. E[X]E[X] = = npnp, , Var(X)Var(X) = = np(1-p)np(1-p)

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Binomial DistributionBinomial Distribution

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

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Negative Binomial DistributionNegative Binomial Distribution

Now, suppose the no. of trials is not fixed.Now, suppose the no. of trials is not fixed. We perform Bernoulli trials repeatedly until We perform Bernoulli trials repeatedly until

a given no. of successes a given no. of successes rr are observed. are observed. Then we stop the trials.Then we stop the trials.

We said the no. of trials required We said the no. of trials required XX is a is a negative binomial random variable:negative binomial random variable:

rnr ppr

nnXP

)1(1

1}{

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Negative Binomial DistributionNegative Binomial Distribution

There are (There are (rr-1) successes in the first (-1) successes in the first (nn-1) -1) trials.trials.

The The nn-th (last) trial must be a success.-th (last) trial must be a success. E[X]E[X] = = rr//pp, , Var(X) = r(1-p)/pVar(X) = r(1-p)/p22

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Negative Binomial DistributionNegative Binomial Distribution

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

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Geometric DistributionGeometric Distribution

It is a particular case of -ve binomial distr.It is a particular case of -ve binomial distr. Suppose we perform Bernoulli trials until a Suppose we perform Bernoulli trials until a

success occurs (i.e. success occurs (i.e. rr=1).=1). If we let If we let YY equal the no. of trials required: equal the no. of trials required:

Then Then YY is a geometric random variable. is a geometric random variable.

111 )1(11-1

1-y=y)=P(Y

yy pppp

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Geometric DistributionGeometric Distribution

Geometric distribution is memoryless:Geometric distribution is memoryless:

Why?Why?)()|( jYPiYjiYP

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Geometric DistributionGeometric Distribution

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