Univariate Probability Distributions

Understand the concept of random variables as a function from sample space to real line.

Understand the concept of probability distribution of a discrete random variable.

Calculation of probabilities for a discrete random variable. Understand the concept of cumulative distribution function.

Introduction

We are introduced to the concept of random experiments, sample space and probability in previous chapters. Many a times, sample space S or 2 contains non-numeric elements. For example, in an experiment of tossing a coin, = {H, T}. However, in practice it is easier to deal with numerical outcomes. In turn, we associate real number with each outcome. For instance, we may call H as 1 and T as 0. Whenever we do this, we are dealing with a function whose domain is the sample space and whose range is the set of real numbers. Such a function is called a random variable (r.v.).

Definition 1: Random variable: Let 2 be a discrete sample space corresponding to a random experiment. A function X : 22 → R (where R is a real line) is called as a random variable.

Remark 1: If 2 = {-4, 0, 4} and X(0)=√o, then X(-4)=√√-4 is an imaginary number.

.. X (-4) R. Hence, X is not a real valued mapping. Hence, X is not a random variable.

Remark 2: If = {a, b, c) and X is a mapping between to A as follows:

Ω = {2,

Fig. 4.1

then X (a) = 0 also X (a) = 1.

i.e. X is a one-many correspondence.

.. X is not a function. Recall that a function can either be one-one correspondence or a many-one correspondence. Hence, X is not a random variable.

Discrete Random Variable

Definition 2: Discrete random variable: A random variable X is said to be discrete if it takes finite or countably infinite number of values. Thus discrete random variable takes only isolated values.

Remark: In this course, we shall deal with only discrete sample spaces containing finite number of elements.

For example,

= {@1, w2,..., n}.

Therefore, in this setup, a discrete random variable is a function X which assigns a real number = X (@) to every sample point ; . Remark : X: {0, 02, 03, ..., n} → A, where A is a finite set of real numbers.

Illustration 1: Suppose two coins are tossed simultaneously. = {HH, TH, HT, TT}

Let X be number of heads obtained tossing two coins. Then we have, X: R in the following manner.

X (HH) = 2, X (TH) = 1, X (HT) = 1, X (TT) = 0.

Here random variable X takes three distinct values 0, 1, 2. Following diagram will help in understanding the concept of random

variable.

Ω = {HH, TH, HT, TT}

Ω = {ΗΗ, ΤΗ, HT, TT}

0 1 Fig. 4.2

Note: Random variables are denoted by capital letters X, Y, Z etc., whereas the values taken by them are denoted by corresponding small letters x, y, z etc. Remark

Several random variables can be defined on the same sample space . For example in the Illustration 1, one can define Y = number of tails or Z = difference between number of heads and number of tails. Following are some of the examples of discrete variable. Number of students present in the class.

(i)Number of accidents on a highway.

(ii) Number of days of rainfalls in Pune city.

(iii) Number of patients cured by using a certain drug.

(iv)Number of attempts required to pass the examination

Definition 3: Range set of a discrete random variable: Let X be a discrete random variable defined on a sample space. Since N contains either finite or countably infinite elements, and X is a function on 2, therefore, X can take either finite or countably infinite values. Suppose X takes values X1, X2,..., then the set {X1, X2, ... } is called the range set of X.

For the illustrative example 1, observe that the range set of X = number of heads is {0, 1, 2}.

Probability Distribution

Consider again the experiment of tossing two unbiased coins. X = number of heads observed in each tossing. Then range set of X={0, 1, 2}. Although, we cannot in advance predict what value X will take, we can certainly state the probabilities with which X will take the three values 0, 1, 2. The following table helps to determine such probabilities.

Outcome

Probability of outcome

Value of X

Observe that the following events are associated with the distinct values of X.

(X=0)

{TT}

(X = 1)

(X = 2)

{TH, HT} {HH}

Therefore, probabilities of various values of X are nothing but the probabilities of the events with which the respective values are associated

P(X=0) = P(TT) =

P(X = 1) = P(TH, HT) =)

P(X = 2) = P(HH) =

Note that these probabilities add upto 1 (why ?).

Definition 4 Probability mass function: Let X be a discrete random variable defined on a sample space . Suppose (X1, X2, ..., Xn} is the range set of X. With each of xi.

We assign a number p;= P(X = x;), called the probability of x; such that

(i)

Pi≥ 0

; i = 1, 2, ..., n

Σ pi = 1

and (ii)

Then the function P defined above is called the probability mass function (p.m.f.) of X.

The table containing the values of X along with their probabilities given by probability mass function is called as probability distribution of the random variable X. For example,

...

P (X = X1)

...

Total 1

Remark 1: Properties of a random variable can be studied only in terms of its p.m.f. We need not refer to the underlying sample space, once we have the probability distribution of the random variable.

Remark 2

P(X = A) =

If A is any subset of the range set of X, then

p (x;); where the sum is taken over all points in A. xje A

For example, if the range set of X is {1, 2, 3, 4, 5, 6} and we want to find P (X ≤ 3), then,

P(X ≤3) P (X = 1) + P(X = 2) + P(X = 3)

= P1+ P2+P3

Illustration 2: A symmetric die is rolled and number on uppermost

face is noted. Find its probability distribution.

Solution: X = number on uppermost face.

Range set of X= (1, 2, 3, 4, 5, 6}

Probability of each of the elements = 6 5.

The probability distribution of X is

Total

P (X = x)

Illustration 3: A pair of fair dice is thrown. Let X = sum of numbers on the uppermost faces.

Since,

Range set of X = {2, 3,..., 12}

X (1, 1) 2 and X (6, 6) = 12.

We know that contains 36 elements (ordered pairs).

The following table displays the subsets corresponding to the events (X=j); j = 2, ..., 12 as well as the corresponding probabilities pj.

Value of X

Subsets of Ω

Pj = P(X=j)

{(1, 1)}

1/36

{(1, 2), (2, 1)}

2/36

{(1, 3), (2, 2), (3, 1)}

3/36

{(1, 4) (2, 3), (3, 2), (4, 1)}

4/36

{(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}

5/36

{(1, 6) (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}

6/36

{(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)}

5/36

{(3, 6), (4, 5), (5, 4), (6, 3)}

4/36

{(4, 6) (5, 5), (6,4)}

{(5, 6) (6,5)}

3/36

2/36

{(6,6)}

1/36

Note that (i) pj≥0, Vj = 2, 3,..., 12 and (ii)

Pi = 1.

Cumulative Distribution Function (c.d.f.) or Distribution Function (d.f.)

Definition 5: Let X be a discrete random variable taking values x1, X2, ..., Xi, ..., Xn with probabilities P1, P2, ..., Pi, ..., Pn respectively. Then cumulative distribution function (c.d.f.) which is also called as distribution function (d.f.) is denoted by F (x) and is defined as follows. F(x) = P(X ≤x]; -∞<x<∞ F(x) = P(X ≤x1]

In particular,

Σ Pi

; i = 1, 2, ..., n.

j=1

Remark: The c.d.f. is defined for all values of xe R. However, since the random variable takes only isolated values, the function is constant in between two successive values of X and have jumps at the points xi=1,2,3,.....n.Hence distribution function of a discrete random variable is a step function.

Properties of Distribution Function F (.)

The distribution function is a very important entity Statistics. It is used extensively in Statistical Inference, the main branch of Statistics; in which inferences are drawn regarding population on the basis of information collected in a sample. Now, we shall study some of the important properties of c.d.f. F (x) of a discrete random variable.

(i) F(x) is defined for all xe R, real line.

(ii)0≤F(x)≤1; obvious, as it is probability of the event (X ≤ x).

(iii) F(x) is a non-decreasing function of x.

i.e. if ab, then F (a) < F (b).

This is clear from the graph.

(iv) It has jumps at X1, X2, ..., Xn, the values taken by the random

variable X and is constant between two successive values of X. Moreover, size of jump at x; is P (X = x;).

(v) F(-) 0 and F (∞) = 1.

where, F(-∞) =

lim X-∞

lim

F(x), F (∞0) =

F(x).

(vi) Let a and b be two real numbers where a < b; then using distribution function, we can compute probabilities of different events as follows.

Median of A Discrete Probability Distribution: Let X be a discrete random variable with c.d.f. F(x). The median M of the probability distribution of X is defined as that value of X, such that

P(X ≤M) > 1/2

and P(X > M) > 1/2

In other words,

F (M) ≥ and 1 -F (M) + P(M). For calculation purposes,

median is the first value of X for which F (X) ≥ 0.5.

Illustration 5: Consider the following probability distribution.

X P (x)

0.1 0.2

0.3 0.25 0.15

The c.d.f. F(x) is given as follows:

X F(x)

Hence, X 3 is the median;

1 0.1

0.3 0.6

4 5 0.85 1

since F(3)

0.6≥ 0.5

M = 3

and 1-F (3) + P(3) = 1-0.6+0.3 0.7 0.5

Illustration 6: Consider the following probability distribution.

X P (x)

1 1

Observe that, Median = 2

P(X ≤2) =

and P(X 2) =

Also,

P(X ≤3) =

and P(X≥ 3) =52

Hence, Median = 3.

Thus median may not be unique.

Mode of a Discrete Probability Distribution

Let X be a discrete random variable with c.d.f. F (x). The mode, Mo of the probability distribution of X is defined as the value of X for which the p.m.f. is maximum.

Illustration Determine mode for the following probability distribution.

X P (x)

10 0.13 0.15 0.24 0.37

0.11

Observe that the Mode = 20.

maximum p.m.f. is 0.37 P (20). Hence,

Remark: The mode may not be unique. If there are more than one values of X, for which the p.m.f. is maximum, then all these values of X are modes. If there are two modes, the distribution is called bimodal. If there are more than two modes, say 3, 4, ..., then the distribution is called multimodal.

Probability Distribution of a Function of a Discrete r.v.

Let X be a discrete r.v. with the following probability distribution.

X2 P2

...

Xn Pn

P (X = x) Pi Suppose, Yg (x) is a real valued function of X. Then Y takes

values

Y1 = g(x1), y2 = g(x2),... Yi = g(xi), .... Yn = g(xn). Hence, the probability distribution of Y will be as follows:

P(Y = y)

y1 = g(x1) P1

...

Yi = g(xi) Pi