In most of the random experiments that we perform, we are not only interested in the particular outcome of the experiment but in some number associated with that outcome. For example: when we roll a pair of dice, we may be interested in the sum of numbers that appears on the dice. In such experiments, a single real number is assigned to each outcome of the experiment. This single real number varies with different outcomes of the experiment. It is called as random variable as its value depends upon the outcome of a random experiment. Thus, a random variable is defined as a real valued function whose domain is the sample space of a random experiment.

Probability distribution function

Probability density function is a function that describes the likelihood of the random variable to take on a given value. The probability of a random variable that falls in a range of values is the integral of the variable’s density. It is given by the area under density function over the horizontal axis and between the highest and lowest values of the range. The probability density function is always positive and its integral above the complete space is one.

Continuous random variable

A continuous random variable in probability is defied as a variable where the data can accept infinite values. For instance, a random variable that calculates the time required for any task is continuous since there are infinite number of possibilities that can be required.

Any continuous random variable whose probability density function is f(x), is given by:

For example: In an experiment of tossing a coin three times in succession, let X denotes the number of tails obtained. Thus, X is a random variable, for each outcome its value can be given as,

Sample space, S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}

X (HHH) = 0, X (HHT) = 1, X (HTH) = 1, X (THH) = 1, X (TTH) = 2, X (THT) = 2, X (HTT) = 2, X (TTT) = 3

Illustration 1: A bag contains 1 white and 2 red balls. One ball is drawn at random and then put back in the box after noting its colour. The process is repeated again. If X denotes the number of white balls recorded in the two draws, describe X.

Solution: Let the balls in the bag be denoted by w, r_{1}, r_{2}. Then, the sample space, S can be given as

S = {r_{1}r_{1}, r_{1 }r_{2}, r_{2}r_{2}, r_{2}r_{1}, r_{1}w, r_{2 }w, w r_{1}, w r_{2}, w w}

X ({r_{1}r_{1}}) = X ({r_{1 }r_{2}}) = X ({r_{2}r_{2}}) = X ({r_{2}r_{1}}) = 0

X ({r_{1}w}) = X({r_{2 }w }) = X ({w r_{1}}) = X ({w r_{2}}) = 1

X ({w w}) = 2

Possible values for the random value X are 0, 1, 2

Probability distribution

For a random variable X, probability distribution is defined as a description accounting the values of the all the random variable related to the outcomes of the experiment along with their corresponding probabilities. Thus probability distribution of a random variable X can be given as:

X: x_{1}, x_{2}, x_{3}, ………, x_{n}

P(X): p_{1}, p_{2}, p_{3}, ………., p_{n}

Where, x_{1}, x_{2}, x_{3}, ………, x_{n} are the different values of random variable X and p_{1}, p_{2}, p_{3}, ………., p_{n} are the probability corresponding to the values of random variable. Thus,

\(\sum\limits_{i = 1}^{n}~p_i~=~1\)

Illustration 2: From a well-shuffled deck of 52 cards, two cards are drawn successively with replacement. Find the probability distribution of the number of kings.

Solution: Let X be a random variable denoting the number of kings. Possible values of X are 0, 1,2.

P(X = 0) = P(non-king and non-king) = P(non-king) × P(non-king)

= \(\frac{48}{52}~\times~\frac{48}{52} ~=~\frac{144}{169}\)

P(X = 1) = P(king and non-king or non-king and king) = P(king and non-king) + P(non-king and king)

= P(king) × P(non-king) + P (non-king) × P(king) = \(\frac{48}{52}~\times~\frac{4}{52}~+~\frac{4}{52}~\times~\frac{48}{52}~ =~\frac{24}{169}\)

P(X = 2)

= P (king and king)= \(\frac{4}{52}~\times~\frac{4}{52}~=~\frac{1}{169}\)

Thus, the probability distribution can be given as,

X | 0 | 1 | 2 |

P(X) | \(\frac{144}{169}\) |
\(\frac{24}{169}\) |
\(\frac{1}{169}\) |

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