random variables

Let us learn about random variables
If is often very important to allocate a numerical value to an outcome of a random experiment. For example consider an experiment of tossing a coin twice and note the number of heads (x) obtained.

Outcome : HH HT TH TT

No. of heads (x) : 2 1 1 0

x is called a random variable, which can assume the values 0, 1 and 2. Thus random variable is a function that associates a real number to each element in the sample space.

Random variable (r.v)

Let S be a sample space associated with a given random experiment.

A real valued function X which assigns to each wi Î S, a unique real number.

X(∞i) = Xi is called a random variable

Note:

There can be several r.v's associated with an experiment.

A random variable which can assume only a finite number of values or countably infinite values is called a discrete random variable.

e.g., Consider a random experiment of tossing three coins simultaneously. Let X denote the number of heads then X is a random variable which can take values 0, 1, 2, 3.

Continuous random variable

A random variable which can assume all possible values between certain limits is called a continuous random variable.

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binomial probability distribution

The binomial probability distribution is known as the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. Such as the success and also the failure experiment is also called a Bernoulli experiment or Bernoulli trial. In fact, when n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis and also the popular binomial test of statistical significance.

Binomial Distribution on Binomial Probability Distribution Table:
Definition of Binomial Distribution :
A random variable X is said to be follow as Binomial distribution if its probability mass function is given by
P(X = x) = p(x) = { nC_x p^x q^{n −x}, x = 0, 1, . . .n
Constants of Binomial Distribution :
Mean = np
Variance = npq
Standard deviation = Variance = npq
X ∼ B(n, p) denotes that the random variable X follows Binomial distribution with parameters n and p

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