Maharashtra Board Class 12 Maths Part 2 Chapter 7 Probability Distributions PDF Download

7. Probability Distributions

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 elements, 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 to 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, therefore it is 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 \(2^{10}\) = 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 outcome 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 set of real numbers.

Thus \(X : S \to \mathbb{R}\) is a random variable.

We often use the abbreviation r.v. to denote a 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 and the letter N when a seed does not germinate. 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 in 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 usually denoted by a capital letter, like X or Y. A particular value taken by the random variable is usually denoted by the small letter x. Note that x is always a real number and the set of all possible outcomes corresponding to a particular value x of X is denoted by the event \([X = x]\).

Teacher's Note

A random variable takes one value for each outcome. For example, if you roll a die, the random variable "number shown" takes value 1, 2, 3, 4, 5, or 6 based on what you get.

Exam Trick

Remember: Random variable = Number from outcome. Just like your mobile phone number identifies you, the random variable identifies each outcome of the experiment with a number.

Points to Remember

A random variable is a function from sample space to real numbers.
It changes its value based on the outcome of the experiment.
We write it as capital letters like X, Y, or Z.
Each outcome gets exactly one value of the random variable.
The set of all possible values is called the range.

Types of Random Variables

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

Discrete Random Variables

A random variable is said to be a discrete random variable if the number of its possible values is finite or countably infinite.

The values of a discrete random variable are usually denoted by non-negative integers, that is, \(\{0, 1, 2, \ldots\}\).

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, number of stars in the sky and so on.

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

Continuous Random Variable

A random variable is said to be a continuous random variable if the possible values of this random variable 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.

Teacher's Note

Discrete variables are things you count, like the number of students in a class. Continuous variables are things you measure, like the height of students in meters.

Exam Trick

Remember: Discrete = Count (1, 2, 3...). Continuous = Measure (1.5 meters, 2.3 kg). If you can list all values, it is discrete. If it fills a range, it is continuous.

Points to Remember

Discrete random variable has countable values like 0, 1, 2, 3.
Continuous random variable has values in an interval like 0 to 1.
Discrete is obtained by counting things.
Continuous is obtained by measuring things.
Both are real-valued functions on the sample space.

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