random variable | random variable examples | continuous random variable

What is a random variable?

Random variable is a variable which represent the outcome of an experiment or a trial or an event. It is specific number which is different each time the trial or experiment or event

is repeated.

Random variableis a measurable function from a sample space, as a set of possible outcomes from the random experiment. Random variable can take on many values,

this is because there can be several outcomes of random experiment. Random variable

should not be confused with an algebraic values. Random variable can have a set

of values that could be the resulting outcome of a random experiment.

Definition of random variable

Random variable can be defined as a type of variable whose value depends up on the numerical outcome of a certain random experiment, it is also known as stochastic variable.Random variable are always a real number as they are required to be measurable.

Examples of random variable;
For example if E consists of two tosses of a coin. we may consider the random variable which is the number of heads(0,1 or2)

Outcome; HH HT TH TT

Value of X; 2 1 1 0

here X is the random variable which represents the number of times the head may occur in the experiment ,that is tossing of a coin, here 2 represents that head appear two times etc

There are two types of random variable

1.Discrete random variable

2. Continuous random varaible

1.Discrete random variable

Discrete random variable is a variable that can take any whole number values as outcomes of a random experiment. Discrete random variable takes countable number of possible outcomes and it can be counted as 0,1,2, 3,......, it is also known as a stochastic variable. Discrete random variables are always whole numbers which are easily countable. probability mass function is used to describe the probability distribution of a discrete random variable

If a random variable takes at most a countable number of values it is called a discrete random variable, in other words a real valued function defined on a discrete sample space is called discrete random variable.

Mean of discrete random variable

The average value of a random variable is called the mean of a discrete random variable the mean is also known as the expected value it is generally denoted by E(X).where X is the random variable ;

mean of a discrete renovial is equal to =$$E(X)=[\sum {xp(X = x)} ]$$ where P(X=x) is the probability mass function of random variable.
Where
$$\sum\limits_{i = 1}^n {p(x) = 1} $$
i=1,2,3,.......

Variance of Discrete random variable

Variance of discrete random variable can be defined as the expected value of the square of the difference of the random variable from the mean,hence mathematically;

\[v(x) = \sum\limits_{i = 1}^n {{{({X_i} - \mu )}^2}} p(X = x)\]

i=1,2,3,.......

Types of discrete random variable are, binomial random variable, geometric random variable, bernoulli random variable.

Example solved;

2. Continuous Random variable

Continuous random variable is a random variable that can take an infinite number of possible values, is known as a continuous random variable. Such a variable is defined over an interval of values rather than a specific value, and example of continuous random variable is the weight of a person.In other words a random variable is said to be continuous, if it is continuous that falls between a particular intervals. Continuous random variables are used to denote measurement such as height, weight, time, etc.

Random variable X is said to be continuous random variable if it can take all possible values between certain limits. In other words we can say that random variable is said to be continuous random variable when its different values cannot be put in 1-1 correspondence with a set of positive integers.

Mean and variance continuous random variable

Mean of continuous random variable

The mean of continuous random variable can be defined as weighted average value of the random variable X. It is also known as the expectation of the continuous random variable ,

mean of continuous random variable=\[E(X) = \mu = \int\limits_{ - \infty }^{ + \infty } {xf(x)dx} \]

Variance of continuous random variable.

Variance of continuous random variable can be defined as the expectation of the squared difference from the mean, the formula is given as follows;

\[V(X) = {\sigma ^2} = \int\limits_{ - \infty }^{ + \infty } {{{(x - \mu )}^2}f(x)dx} \]

uniform random variable, exponential random variable, normal random variable , standard normal random variable are examples of continuous random variables.

Example solved;