Probability Distributions

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Probability Distributions

• www.ibmaths.com

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When using this slideshow, ensure that you have a
copy of pages 11 and 12 of the IB HL information
booklet. This gives the notation, probability
mass functions, mean and variance. The formal
proofs of the means and variances of these
distributions is not required.
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Bernoulli
The Bernoulli distribution refers to a trial
where there exists either a success or a
failure. This distribution was developed by the
Swiss mathematician Jacob Bernoulli in the late
17th century and is the basis of the binomial
distribution.
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Binomial distribution
The binomial distribution is covered in the core
part of the HL course. It is a distribution that
has two outcomes classified as either a success
or a failure and a fixed number of trials. It
is an extension of the Bernoulli distribution.
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Geometric
The geometric distribution is a further
application of the Bernoulli. It is a
distribution where a trial is repeated until the
first success occurs.
Example A die is rolled until a 6 is obtained.
Let X be the number of trials required before a
success is obtained. This is a table of
probabilities.
Use your table of formulae to find the mean and
variance of this distribution.
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Negative Binomial distribution (Pascals)
The negative binomial is used to give r successes
in x trials, with the final success being on the
final trial. The probability of a success is
given by p, and a failure as q.
Example An biased die is rolled until 2 sixes
have been obtained. Find the probability that 5
rolls will be needed.
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Hypergeometric
The hypergeometric distribution occurs when a
sample is taken (n), without replacement, from a
population (N). The number of successes in the
population is given as M. The number of successes
in the sample is given by x. The probability can
be calculated as .
Example 10 cards are chosen from a pack of
cards. Find the probability that exactly 1 of the
10 cards are aces.
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Discrete uniform
This is a simple distribution where all the
outcomes are equally likely to occur. A good
example of this is the probability of obtaining a
six when an unbiased die is rolled. This
distribution is sometimes known as the equally
likely outcomes.
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Continuous uniform
This is used for continuous data that is spread
equally, sometimes known as the rectangular
distribution. It has two parameters, a and b.
The probability densities are equal.
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Exponential
The exponential distribution is used to show
naturally occurring probability models thats
could include -half life radioactive
models -life span of electronic devices -length
of time between successive accidents in a large
factory
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Exponential distribution
Referring back to the core part of the
probability course for HL we can get the mean and
variance of the continuous probability
distributions as follows
Mean
This simplifies to give
Variance
This simplifies to give
The proofs of these are not required.
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Other distributions
In addition to these distributions you will need
to be aware of the distributions from the core
part of the course Poisson distribution and the
normal distribution.