A random variable is a real valued function whose domain becomes the complete sample space of a random experiment.

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Random variables can be divided into two parts: Discrete random variable and Continuous random variable.

A discrete random variable can take only a finite number of different values like 0, 1, 2, 3, 4, ..and so on, whereas a continuous random variable is a variable that can take an infinite number of possible values in a particular interval.

Concept of Random Variables includes the following important definitions and formulas:

1. The probability distribution of a random variable X is the system of numbers

X : x_{1}, x_{2}, ..., x_{n}

P(X) : p_{1}, p_{2}, ... , p_{n}

where, p_{i} > 0, and i varying from 1 to n, p_{i} = 1, i = 1, 2, ..., n.

2. The probabilities p_{i}, where P(X = x_{i}) = p_{i}, must satisfy the following:

1: 0 < p_{i} < 1 for each i

2: p_{1} + p_{2} + ... + p_{k} = 1.

3. Let X be a random variable with x_{1}, x_{2}, x_{3}, ..., x_{n} as the possible values and p_{1}, p_{2}, p_{3}, ..., p_{n} as the corresponding probabilities respectively. The mean of X, denoted by μ,

is the number ∑ x_{i} p_{i}, i = 1, 2,…,n. Mean of a random variable X is known as expectation of X, denoted by E (X).

4. Let X be a random variable having values x_{1}, x_{2}, ..., x_{n} occur with probabilities p(x_{1}), p(x_{2}), ..., p(x_{n}) respectively.

Let μ = E(X) be the mean of X. The variance of X, denoted by Var (X) or σ_{x}^{2}, is defined as σ_{x}^{2} = Var (X) or equivalently σ_{x}^{2} = E (X – μ)^{2}

5. The non-negative number σ_{x} = sqrt (Var (X)) is called the standard deviation of the random variable X.

6. The formula for variance: Var (X) = E (X^{2}) – [E(X)]^{2}

7. The Trials of a random experiment will be known as Bernoulli trials, if they will satisfy the following points:

(i) The trials should be finite in numbers and should be independent.

(ii) Each trial can have 2 possibilities either success or failure.

(iii) In each trial, the probability of success remains equal.

Solution: Here, the random variable X will be the number of tails that can be noted down.

Therefore, X can only take the values 0, 1, ..., 10, so X is a discrete random variable.

Therefore, X can only take the values 0, 1, ..., 10, so X is a discrete random variable.

Solution: Here, random variable X denotes the lifetime in hours and hence X could take any positive real value, so X is a continuous random variable.

Solution: Both event are equally likely: (X = 1) = (X = 0) = 1 /2 and hence, it’s a case of Bernoulli trials.

More topics in Random Variable | |

Discrete Random Variable | Continuous Random Variable |