BinomialDistribution - Maple Help

The Binomial Distribution

Main Concept

The binomial distribution is a discrete probability distribution that is used to obtain the probability of observing exactly k number of successes in a sequence of n trials, with the probability of success for all single trials of p. The binomial distribution describes a distribution where there are two mutually exclusive outcomes to an event. When n = 1, the binomial distribution is a Bernoulli distribution.



 If a random variable X follows the binomial distribution, the probability of getting k successes in n trials is given by the following probability mass function:       for where:     p is probability of a successful event k is the number of success trials n is the number of trials The cumulative distribution function is defined as:       for   Note that the probability function and cumulative distribution function for the binomial distribution are only defined for integer values for k and as such, there is no continual curve which can be drawn through the point.  The gray lines are only for illustrating the shape of each function over the interval.

Properties

If n = number of trials, and p = probability of a successful event then:

 PMF The probability mass function CDF The cumulative distribution function Mean $\mathrm{np}$ The expected value of a binomial random variable Standard Deviation $\sqrt{\mathrm{np}\left(1-p\right)}$ Represented by the symbol σ, representing how much variation or spread exists from the mean value.

 Example Suppose a biased coin comes up head with a probability of 0.2 when tossed. What is the probability of achieving 0, 1, 2, 3,and 4 heads after 4 tosses?   Let probability of success, p = 0.2, and number of trials, n = 4 

See how the probability function of the binomial distribution changes with different values for the number of successes k, number of trials n, and the probability of success p.

 # of successes (k) = # of trials (n) = Probability of success (p) =



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