1. Some Discrete Probability Distributions
Discrete Uniform Distribution
We roll a fair die.
Let X = the # that comes up.
This is an example of equiprobable outcomes, that is
To state the probability distribution of X we need to give its possible values and its pmf
X is a discrete Uniform random variable. X has a uniform distribution.
In general, the discrete Uniform distribution is given by…
The mean and variance of the discrete Uniform distribution are…
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2. Bernoulli Distribution
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3. Binomial Distribution
Roll a die n time and count the number of times 6 came up. Let X be the number of 6’s in n rolls. X has image {1, 2, …, n}
The probability distribution of X is given by the following formula
In general, if identical Bernoulli trail is repeated n times independently and X is a random variable that count the number of success in the n trails then the probability distribution of X is given by
Where p is the probability of success on any one experiment.
X is a Binomial random variable. X has a Binomial Distribution.
Question: is this a valid pmf? Prove!
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4. Geometric Distribution
We roll a fair die until the first 6 comes up.
Let X = the number of rolls until we get the first 6.
Possible values of X: {1, 2, 3, …..}
The probability distribution of X is given by the following formula
In general, if identical Bernoulli trail is repeated independently until the first success is obtained and X is a random variable that count the number of trials until the first success then the probability distribution of X is given by
X is a Geometric random variable. X has a Geometric Distribution.
Question: is this a valid pmf? Prove!
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5. In general for a Geometric distribution:
Memory-less property of geometric random variable: for i > j
The mean and variance of the geometric distribution are…
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6. Negative Binomial Distribution
We roll a fair die until the second 6 comes up. This is the waiting time for the second 6. Let X = the number of rolls until we get two 6’s.
Possible values of X: {2, 3, 4, …..}
The probability distribution of X is given by the following formula
Is this a valid pmf? Prove!
In general, X is the total number of experiments when waiting for rth success in a sequence of independent Bernoulli trails. The probability distribution of X is given by
X has a Negative Binomial random Distribution.
The mean and variance are ….
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7. Hypergeometric Distribution
A hat contains 12 tickets, 7 black and 5 white. Three tickets are drawn at random.
Let X = the # of black tickets drawn. X could be 0, 1, 2, 3.
The probability mass for each value can be calculated using combinatorics. For example,
In general, the probability distribution of the hypergeometric random variable is given by
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8. Poisson Distribution
Model for the number of events occurring in a time (or space) interval where λ (a parameter of the distribution) is the rate of the occurrence of the events per one unit of time (or space).
A Poisson random variable X = number of events per one unit of time (space).
Possible values for X: {0, 1, 2, … }
The probability distribution of X is given by
Is this a valid pmf? Prove!
The mean and variance are…
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9. Exercises
A box contain 20 notes numbered 20 to 39. We randomly pick one note and record its number. What is the probability that the number we got is greater then 32?
30% of U of T students wear glasses. We select a random sample of size 10 students.
a) What is the probability that exactly 4 of them wear glasses?
b) What is the probability that more then 3 wear glasses?
We roll a die until we obtained an even outcome.
a) What is the probability that we will roll the die exactly 5 times?
b) What is the probability that we roll the die more then 7 times ?
c) What is the probability that we roll the die more then 7 times if we know that we need more then 2 rolls?
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10. We roll a die until we get 6 even outcomes.
a) What is the probability that we need exactly 10 rolls?
b) What is the probability that we need less 10 rolls?
The number of cars that cross Spadina and Bloor intersection is a Poisson random variable with λ = 15 cars per minute.
a) What is the probability that in a given minute exactly15 cars will cross the intersection?
b) What is the probability that in a given minute more then 15 cars will cross the intersection?
c) What is the probability that during half an hour there where exactly 2 minutes in which 15 cars crossed the intersection?
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11. Relation between Binomial and Poisson Distributions
Binomial distribution
Model for number of success in n trails where P(success in any one trail) = p.
Poisson distribution is used to model rare occurrences that occur on average at rate λ per time interval. Can think of “rare” occurrence in terms of p  0 and n  ∞. Take these limits so that λ = np.
So we have that
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