Module 3. Probability distributions
Lesson 9
RANDOM VARIABLE AND ITS PROBABILITY DISTRIBUTION
9.1 Introduction
In this lesson we will study the important concept of random variable and its probability distribution. It has been a general notion that if an experiment or a trial is conducted under identical conditions, values so obtained would be similar. But this is not always true. The observations are recorded about a factor or character under study e.g. fat, SNF, TS, moisture content etc. in a dairy product. These can take different values, and the factor or character is termed as variable. These observations vary even though the experiment has been conducted under identical conditions. Therefore, we have a set of results or outcomes (Sample points) of a random experiment. A rule that assigns a real number to each outcome (Sample points) is called a random variable. A random variable is a rule or function that assigns numerical values to observations or measurements. It is called random variable because the number that is assigned to the observation is a numerical value which varies randomly. A random variable takes a numerical value with some probability. Hence,, a list of values of a random variable together with their corresponding probabilities of occurrence is termed as probability distribution. We shall also find mean and variance of a probability distribution.
9.2 Random Variable
A
random variable (r.v.) is defined as a real number X connected with the outcome
of a random experiment E. For example, if E consists of three tosses of a coin,
we may consider random variable X which denotes the number of heads (0, 1, 2 or
3)
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Outcome : |
HHH |
HTH |
THH |
THH |
HTT |
THT |
TTH |
TTT |
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Value of X : |
3 |
2 |
2 |
2 |
1 |
1 |
1 |
0 |
Thus, to every outcome there corresponds a real number X(w). Since the points of the sample space corresponds to outcomes, this means that a real number, which we denote by X(w), is defined for each w∈S and let us denote them by w1, w2, ---,w8 i.e. X(w1) = 3, X(w2)=2---, X (w8 ) =0. Thus,, we define a random variable as a real valued function whose domain is the sample space associated with a random experiment and range is the real line. Generally it is denoted by capital letters X,Y, Z, --- etc.
9.2.1 Discrete random variable
If a random variable X assumes only a finite or countable set of values, it is called a discrete random variable. In other words, a real valued function defined on a discrete sample space is called a discrete random variable. In case of discrete random variable we usually talk of values at a point. Generally it represents counted data. For example, number of defective milk pouches in a milk plant, number of accidents taking place in milk plant, number of students in a class, number of milch animals in a herd etc.
9.2.2 Continuous random variable
A random variable is said to be continuous if it can assume infinite and uncountable set of values. A continuous random variable is in which different values cannot be put in one to one correspondence with a set of positive integers. For example, weight of calf at the age of six months might take any possible value in the interval of 160 kg to 260 kg, say 189 kg or 189.4356 kg; likewise milk yield of cows in a herd etc. In case of continuous random variable we usually talk of values in a particular interval. Continuous random variables represent measured data.
9.3 Probability Distribution of a Random Variable
The concept of probability distribution is equivalent to the frequency distribution. It depicts how total probability of one is distributed among various values which a random variable can take.
9.3.1 Probability Mass Function (Discrete Random Variable)
Suppose X is a one-dimensional discrete random variable taking at most a countable infinite number of values X1, X2, ..with each possible outcome X; we associate a number pi = P (X=Xi) = P(Xi) called the probability of Xi, the numbers P (Xi), i=1, 2, must satisfy the following conditions:
a) pi = P (X=Xi) = P (Xi) ≥ 0 i.e. pis are all non-negative.
This function pi = P (X=Xi) or p(x) is called the probability function or probability mass function (p.m.f.) of the random variable X and set of all possible ordered pairs {x, p(x)} is called the probability distribution of the random variable X.
9.3.2 Probability density function (Continuous random variable)
In case of a continuous random variable
X, we talk of probability in an interval. If f(x) is a continuous function of
X, f(x) dx gives the probability that the random variable X, takes value in a
small interval of magnitude dx i.e. 
, then f(x) is called the probability density
function (p.d.f.) of a random variable X. It is also known as frequency
function because it also gives the proportion of units lying in the
interval .
If x has range 
.The
following examples illustrate the concept of probability mass function
Example 1. Two cards are drawn one by one without replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of aces.
Solution: Let X be the random variable, which is the number of aces.
Here X takes values 0, 1, 2
Hence, the probability distribution
is
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X: |
0 |
1 |
2 |
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P (X): |
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Example 2. Four defective milk pouches are accidently mixed with sixteen good ones and by looking at them it is not possible to differentiate between them. Three milk pouches are drawn at random from the lot. Find the probability distribution of X, the number of defective milk pouches.
Solution:Let X be the random variable, which is the number of defective milk pouches.
Here X takes values 0, 1, 2, 3.
Total number of milk pouches = 4 + 16 = 20
Number of defective milk pouches = 4
Hence, the probability distribution is
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X: |
0 |
1 |
2 |
3 |
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P (X): |
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9.4 Mean and Variance of a Random variable
Let X denotes the random variable which assumes values x1,x2,---,xn with corresponding probabilities p1,p2,---,pn .Then the probability distribution be as follow:
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X: |
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P(X): |
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Then
The mean (΅) of the above probability distribution is defined as:
The variance (σ2) is defined as:
Mean of a random variable X is also known as expected value and is denoted by E(X)
Example 3. Find the mean and variance of the number of heads in two tosses of a coin.
Solution: Let X denotes the number of heads obtained in two tosses of a coin. Thus, X takes the values 0, 1, 2.
Now p, the probability of getting a head =1/2 and q, the probability of not getting a head = 1- 1/2 = 1/2
Thus, we have:
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0 |
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0 |
0 |
0 |
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1 |
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1 |
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2 |
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4 |
1 |
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Total |
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1 |
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Example 4. A die is tossed twice. Getting a number greater than 4 is considered a success. Find the variance of the probability distribution of the number of success.
Solution:
Here p, probability of a number greater than 4=2/6=1/3 and q, probability of a
number not greater than 
Thus, we have:
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0 |
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0 |
0 |
0 |
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1 |
4/9 |
4/9 |
1 |
4/9 |
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2 |
1/9 |
2/9 |
4 |
4/9 |
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Total |
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6/9 |
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8/9 |
Hence, the mean 
and the variance
9.5 Mathematical Expectation
Let X denotes a discrete random variable which assumes values x1,x2,---,xn with corresponding probabilities p1,p2,---,pn where p1+p2+---+pn = 1, the mathematical expectation of X or simply the expectation of X, denoted by E(X) is defined as:
i.e. it is sum of the product of different possible values of x and the corresponding probabilities. Hence, mathematical expectation of a random variable is equal to its arithmetic mean.
9.5.1 Some results on expectation
· E ( c ) = c , where c is a constant.
· E (cX) =c E(X) , where c is a constant .
· E (aX+b) =a E(X)+b , where a and b are constants.
· Addition law of expectation: If X and Y are random variables then E(X+Y)=E(X)+E(Y) i.e. expected value of the sum of two random variables is equal to sum of their expected values.
· Multiplication law of expectation: If X and Y are independent random variables then
E(X.Y)=E(X).E(Y) i.e. expected value of the product of two random variables is equal to product of their expected values.
· Variance in terms of expectation:
