This GSEB Class 12 Commerce Statistics Notes Part 2 Chapter 2 Random Variable and Discrete Probability Distribution Posting covers all the important topics and concepts as mentioned in the chapter.

## Random Variable and Discrete Probability Distribution Class 12 GSEB Notes

**Random Variable:**

Let U be a sample space of a random experiment. A function which associates a real number with each outcome of U, is called a random variable and is denoted by symbol X. Symbolically, it is also represented by X: U → R.

1. Discrete Random Variable:

A random variable X, which is capable of assuming all the values in a set of real numbers or its subsets, is called a discrete random variable. Examples of a discrete random variable are: number of children per family, getting number of heads when a coin is tossed three times.

2. Continuous Random Variable:

If a random variable X is an element of R or its subset, whose interval is (a, b), where a < b and is capable of assuming any value in the interval, then that random variable X is called a continuous random variable. Examples of continuous random variable are: age of a person, temperature of a place, production of a company.

**Discrete Probability Distribution:**

Suppose, probability of a discrete random variable X is P[X = x_{i}] = p(xi). If for i = 1, 2, 3, …, n; p(x_{i}) > 0 and \(\sum_{i=1}^{n} p\left(x_{i}\right)\) = 1, then set of real values {p(x_{1}), p(x_{2}), …. p(x_{n})} is called probability distribution of a discrete random variable X.

In tabular form, it is written as under:

Here, 0 < p(x_{i}) < 1 and i = 1, 2, 3, …, n

**Mean and Variance of a Discrete Random Variable:**

Suppose X is a discrete random variable that assumes any one value from x_{1}, x_{2}, ……… x_{n} and the probability distribution of X is as follows:

Here, 0 < p(x_{i}) < 1 and i = 1, 2, 3, …, n

Mean (p) and variance (σ^{2}) are also called the mean and variance of probability distribution of X.

**Binomial Probability Distribution:**

**Law of Dichotomy:**

If possible outcomes of an experiment are only two, the rule is called law of dichotomy.

- Success: If outcome of an experiment is certain, it is called success.
- Failure: If outcome of an experiment is not certain, it is called failure.

**Dichotomous Experiment:**

An experiment, which has only two outcomes, namely Success or Failure, is called a dichotomous experiment.

**Bernoulli Trials:**

If a dichotomous experiment is repeated n times and in each trial probability of success (S) p(0 < p < 1) is constant, then all such trials are called Bernoulli trials.

**Binomial Random Variable:**

If in the series of successes (S) and failures (F) obtained in n numbers of Bernoulli trials, the number of successes (S) is denoted by X, then X is called binomial random variable. X is capable of taking any value of a finite set {0, 1, 2, …, n}.

**Binomial Probability Distribution:**

If probability distribution of x, a value of a binomial random variable X is P(X = x) = p(x) = ^{n}C_{x} pxq^{n-x}

Where, n = Number of Bernoulli trials; x = 0, 1, 2, …, n; 0 <p < 1 and p + q = 1, then it is called binomial probability distribution.

- n and p are parameters of binomial probability distribution.
- Mean of binomial probability distribution = np and Variance of binomial probability distribution = npq
- If the probability of x successes in N trials of a random experiment is p(x), then the expected frequency of x successes is = N – p(x).

**Properties of Binomial Distribution;**

- The mean np of the binomial distribution shows the expected number (average) of successes in n Bernoulli trials.
- The variance of the distribution is = npq. So its standard deviation is = \(\sqrt{n p q}\)
- In the distribution np > npq.
- Probability of failure q = \(\frac{n p q}{n p}\)
- If p < \(\frac{1}{2}\), skewness of the distribution is positive.
- If p > \(\frac{1}{2}\), skewness of the distribution is

negative. - If p = \(\frac{1}{2}\), skewness of the distribution is zero and it becomes symmetrical distribution.

**Symbols:**

- X = Random variable
- x = Value of random variable X
- P (X = x) = Probability of x, a value of X
- n = Number of Bernoulli trials
- x = Number of successes
- p = Probability of success
- q = Probability of failure = 1 -p

**Formulae:
**1. Mean of Random Variable X:

μ = E(X) = Σx . p(x)

2. Variance of Random Variable X:

σ2 = V(X) = E(X – μ)^{2}

= E (X^{2}) – [E(X)]^{2}

= Σx^{2}p(x)- [Σx – p(x)]^{2}

3. Binomial Probability Distribution:

P(X = x) = p(x) = ^{n}C_{x} px q^{n-x} Where, n = Number of Bernoulli trials; x = 0, 1, 2 …… n

p – Probability of success (0 <p < 1)

q = 1 – p