Maharashtra Board Class 12 Maths Commerce Part II Chapter 8 Probability Distributions PDF Download

Probability Distributions

Let's Study

Random variables

Types of random variables

Probability distribution of a random variable

Discrete random variable

Probability mass function

Cumulative distribution function

Expected value and variance

Continuous random variable

Probability density function

Cumulative distribution function

Binomial distribution

Bernoulli trial

Mean and variance of Binomial distribution

Poisson distribution

Let's Recall

A random experiment and all possible outcomes of an experiment.

The sample space of a random experiment.

Let's Learn

8.1 Random Variables

We have already studied random experiments and sample spaces corresponding to random experiments. As an example, consider the experiment of tossing two fair coins. The sample space corresponding to this experiment contains four points, namely {HH, HT, TH, TT}. We have already learnt to construct the sample space of any random experiment. However, the interest is not always in a random experiment and its sample space. We are often not interested in the outcomes of a random experiment, but only in some number obtained from the outcome. For example, in case of the experiment of tossing two fair coins, our interest may be only in the number of heads when two coins are tossed. In general, it is possible to associate a unique real number with every possible outcome of a random experiment. The number obtained from an outcome of a random experiment can take different values for different outcomes. This is why such a number is a variable. The value of this variable depends on the outcome of the random experiment, and is therefore called a random variable. A random variable is usually denoted by capital letters like X, Y, Z, .......

Consider the following examples to understand the concept of random variables.

When we throw two dice, there are 36 possible outcomes, but if we are interested in the sum of the numbers on the two dice, then there are only 11 different possible values, from 2 to 12.

If we toss a coin 10 times, then there are 2¹⁰ = 1024 possible outcomes, but if we are interested in the number of heads among the 10 tosses of the coin, then there are only 11 different possible values, from 0 to 10.

In the experiment of randomly selecting four items from a lot of 20 items that contains 6 defective items, the interest is in the number of defective items among the selected four items. In this case, there are only 5 different possible outcomes, from 0 to 4.

In all the above examples, there is a rule to assign a unique value to every possible outcomes of the random experiment. Since this number can change from one outcome to another, it is a variable. Also, since this number is obtained from outcomes of a random experiment, it is called a random variable.

A random variable is formally defined as follows:

Definition: A random variable is a real-valued function defined on the sample space of a random experiment. In other words, the domain of a random variable is the sample space of a random experiment, while its co-domain is the real line.

Thus X : S → R is a random variable.

We often use the abbreviation r.v. for random variable.

Consider an experiment where three seeds are sown in order to find how many of them germinate. Every seed will either germinate or will not germinate. Let us use the letter Y when a seed germinates. The sample space of this experiment can then be written as S = {YYY, YYN, YNY, NYY, YNN, NYN, NNY, NNN} and n(S) = 8.

None of these outcomes is a number. We shall try to represent every outcome by a number. Consider the number of times the letter Y appears is a possible outcome and denote it by X. Then, we have X(YYY) = 3, X(YYN) = X(YNY) = X(NYY) = 2, X(YNN) = X(NYN) = X(NNY) = 1, X(NNN) = 0.

The variable X has four possible values, namely 0, 1, 2 and 3. The set of possible values of X is called the range of X. Thus, in this example, the range of X is the set {0, 1, 2, 3}.

A random variable is denoted by a capital letter, like X and Y. A particular value taken by the random variable is denoted by the small letter x. Note that x is a real number and the set of all possible outcomes corresponding to a particular value x of X is denoted by the event [X = x]. For example, in the experiment of three seeds, the random variable X has four possible values, namely 0, 1, 2, 3. The four events are then defined as follows:

[X = 0] = {NNN},

[X = 1] = {YNN, NYN, NNY},

[X = 2] = {YYN, YNY, NYY},

[X = 3] = {YYY}.

Note that the sample space in this experiment is finite and so is the random variable defined on it. A sample space need not be finite. Consider, for example, the experiment of tossing a coin until a head is obtained. The sample space for this experiment is S = {H, TH, TTH, TTTH, ....}. Note that S contains an unending sequence of tosses required to get a head. Here, S is countably infinite. The random variable X : S → R, denoting the number of tosses required to get a head, has the range {1, 2, 3, ......} which is also countably infinite.

8.2 Types of Random Variables

There are two types of random variables, namely discrete and continuous.

8.2.1 Discrete Random Variable

Definition: A random variable is a discrete random variable if its possible values form a countable set, which may be finite or infinite.

The values of a discrete random variable are usually denoted by non-negative integers, that is, 0, 1, 2, ......... . Examples of discrete random variables include the number of children in a family, the number of patients in a hospital ward, the number of cars sold by a dealer, and so on.

Note: The values of a discrete random variable are obtained by counting.

8.2.2 Continuous Random Variable

Definition: A random variable is a continuous random variable if its possible values form an interval of real numbers.

A continuous random variable has uncountably infinite possible values and these values form an interval of real numbers. Examples of continuous random variables include heights of trees in a forest, weights of students in a class, daily temperature of a city, speed of a vehicle, and so on.

The value of a continuous random variable is obtained by measurement. This value can be measured to any degree of accuracy, depending on the unit of measurement. This measurement can be represented by a point in an interval of real numbers.

The purpose of defining a random variable is to study its properties. The most important property of a random variable is its probability distribution. Many other properties of a random variable are obtained with help of its probability distribution. We shall now learn the probability distribution of a random variable. We shall first learn the probability distribution of a discrete random variable, and then learn the probability distribution of a continuous random variable.

8.3 Probability Distribution of a Discrete Random Variable

Let us consider the experiment of throwing two dice and noting the numbers on the uppermost faces of the two dice. The sample space of this experiment is S = {(1,1), (1,2), .........., (6,6)} and n(S) = 36.

Let X denote the sum of the two numbers in a single throw. Then the set of possible values of X is {2, 3, ........ , 12}. Further,

[X = 2] = {(1,1)},

[X = 3] = {(1,2), (2,1)},

[X = 12] = {(6,6)}

Next, all of the 36 possible outcomes are equally likely if the two dice are fair. That is, each of the six faces has the same probability of being uppermost when a die is thrown.

As the result, each of these 36 possible outcomes has the probability \(\frac{1}{36}\).

This leads to the following results.

\(P[X = 2] = P\{(1,1)\} = \frac{1}{36}\)

\(P[X = 3] = P\{(1,2), (2,1)\} = \frac{2}{36}\)

\(P[X = 4] = P\{(1,3), (2,2),(3,1)\} = \frac{3}{36}\), and so on.

The following table shows the probabilities of all possible values of X.

Such a description of the possible values of a random variable X along with corresponding probabilities is called the probability distribution of the random variable X.

In general, the probability distribution of a discrete random variable X is defined as follows:

Definition: The probability distribution of a discrete random variable X is defined by the following system. Let the possible values of X be denoted by x₁, x₂, x₃, ........, and the corresponding probabilities be denoted by p₁, p₂, p₃, .... . Then, the set of ordered pairs {(x₁, p₁), (x₂, p₂), (x₃, p₃), ........} is called the probability distribution of the random variable X.

For example, consider the coin-tossing experiment where the random variable X is defined as the number of tosses required to get a head. Let the probability of getting head be t and that of not getting head be 1 − t. The possible values of X are given by the set of natural numbers, {1, 2, 3, .....} and \(P[X = i] = (1 − t)^{i−1}t\), for i = 1, 2, 3, ..... This result can be verified by noting that if head is obtained for the first time on the ith toss, then the first i − 1 tosses have resulted in tail. In other words, [X = i] represents the event of having i − 1 tails followed by the first head on the ith toss.

Teacher's Note

Random variables help us count and measure things from real life. Like counting how many students pass a test or measuring the height of students in your class. It makes difficult math easier.

Exam Trick

Remember: Discrete = Counting (like number of heads). Continuous = Measuring (like height or weight). Look for the word "number" for discrete!

Points to Remember

A random variable is a function that changes based on random outcomes.

Discrete random variables give countable values like 0, 1, 2, 3.

Continuous random variables give any value in a range like 1.5 to 2.8.

The sample space is all possible outcomes. The range is all possible values of the random variable.

Capital letters like X show the random variable. Small letters like x show one value.

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