Distribution Function:
Let X be a random variable. The function F defined for all real x by
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is called the distribution function (d.f.) of the random variable (X).
Discrete Uniform Distribution:
A random variable X is said to have a discrete uniform distribution over the range [1, n] if its probability mass function is expressed as follows.
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Here n is known as the parameter of the distribution and lies in the set of all positive integers. It is also called a discrete rectangular distribution. These distributions can be conceived in practice if under the given experimental conditions, the different values of the random variable become equally likely. For a die experiment, deck of cards experiment.
Moments of Uniform Distribution:
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The moment generating function of X is.
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Bernoulli Distribution:
A random variable is said to be a Bernoulli distribution with parameter p if its probability mass function is given by.
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The parameters p satisfies 0 less than or equals to p less than or equals to 1. Often (1-p) is denoted as q.
A random experiment whose outcomes are of two types, success S and failure F, occurring with probabilities p and q respectively, is called a Bernoulli trail. If for this experiment, a random variable. X is defined such that it takes value 1 when S occurs and 0 if F occurs, then X follows a Bernoulli distribution.
Moments of Bernoulli Distribution:
The rth moment of origin is:
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The moment generating function of Bernoulli variate is:
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Binomial Distribution:
A random variable X is said to be follow binomial distribution if it assumes only non-negative values and its probability mass function is given by:
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The two independent constants n and p in the distribution are known as the parameters of the distribution. n is also sometimes as degree of binomial distribution. It is discrete distribution as X can take only the integral values as 0,1, 2, ......, n. Any random variable which follows binomial is known as binomial variate. The notation X ~ B (n, p) to denote the random variable with the parameters n and p.
Mean and Variance of Binomial distribution:
we know that the probability mass function of binomial distribution is.
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By the definition of expectation mean
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Similarly,
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Recurrence relation of raw moments for binomial distribution:
We know that.
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Additive property or reproductive property:
Statement: If X1, X2, .... , Xn are the 'm' independent random variables which are assumed to follow binomial distribution with the parameters (n1, p1), (n2, p2), ...... , (nm, pm) then the sum of three variables. The sum of Xi for every value of i from 1 to n does not follow binomial distribution.
Proof:
Given that.
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and so, on
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By the additive property of moment generating function is.
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Since, we cannot combine the terms we conclude that the sum of Xi for every value of i from 1 to n does not follow binomial distribution.
If,
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Then, the sum of Xi for every value of i from 1 to n follows the binomial distribution.
Properties of Binomial distribution:
- Mean of binomial distribution is np and variance are npq. Also mean is always greater than variance.
- Moment generating function of binomial distribution is.
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- Characteristic function of binomial distribution is.
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- Probability generating function of binomial distribution is.
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- Binomial distribution does not satisfy the additive property.
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Problems:
1. Find the binomial distribution!
a) If mean = 4 and variance = 3
Given that, np=4 and npq = 3
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b) If mean = 3 and variance = 4
We know that for binomial distribution mean is greater than variance. Since, the mean is less than variance the data is inconsistent.
2. Find the mean, variance and mode for the moment generating function of binomial distribution.
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We know that the moment generating function of binomial distribution.
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3. If mean = 16 and variance = 8 then find the binomial distribution and mode.
Given that,
np = 16 and npq = 8
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The binomial distribution is.
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4. If n = 6 and 9p(x=4) = p(x=2) then find the binomial distribution also calculate p(x=1).
Given that
n = 6 and 9p(x = 4) = p (x = 2)
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P = - 1/2 is invalid because it denotes the probability of success.
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