Theoretical Distributions
Theoretical Distributions
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Theoretical distributions are
Discrete Probability distribution Bernoulli distribution
- A random variable x takes two values 0 and 1, with probabilities q and p ie., p(x=1) = p and p(x=0)=q, q-1-p is called a Bernoulli variate and is said to be Bernoulli distribution where p and q are probability of success and failure. It was given by Swiss mathematician James Bernoulli (1654-1705)
Example
- Tossing a coin(head or tail)
- Germination of seed(germinate or not)
Binomial distribution
- Binomial distribution was discovered by James Bernoulli(1654-1705). Let a random experiment be performed repeatedly and the occurrence of an event in a trial be called as success and its non-occurrence is failure. Consider a set of n independent trails( n being finite), in which the probability p of success in any trail is constant for each trial. Then q=1-p is the probability of failure in any trail.
- The probability of x success and consequently n-x failures in n independent trails. But x successes in n trails can occur in ncx ways. Probability for each of these ways is pxqn-x.
P(sss…ff…fsf…f)=p(s)p(s)….p(f)p(f)…. =p,p…q,q… =(p,p…p)(q,q…q) (x times) (n-x times)
Hence the probability of x success in n trials is given by
ncx pxqn-x
Definition
- A random variable x is said to follow binomial distribution if it assumes non-negative values and its probability mass function is given by
The two independent constants n and p in the distribution are known as the parameters of the distribution.
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Last modified: Monday, 19 March 2012, 7:22 PM