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 = xi] = p(xi). If for i = 1, 2, 3, …, n; p(xi) > 0 and \(\sum_{i=1}^{n} p\left(x_{i}\right)\) = 1, then set of real values {p(x1), p(x2), …. p(xn)} is called probability distribution of a discrete random variable X.
In tabular form, it is written as under:
Here, 0 < p(xi) < 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 x1, x2, ……… xn and the probability distribution of X is as follows:
Here, 0 < p(xi) < 1 and i = 1, 2, 3, …, n
Mean (p) and variance (σ2) 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) = nCx pxqn-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 (X2) – [E(X)]2
= Σx2p(x)- [Σx – p(x)]2
3. Binomial Probability Distribution:
P(X = x) = p(x) = nCx px qn-x Where, n = Number of Bernoulli trials; x = 0, 1, 2 …… n
p – Probability of success (0 <p < 1)
q = 1 – p