Chapter 4: Random Variable
Definition: A function whose domain is the set of possible outcomes, and whose range is a subset of the set of reals. Such a function is called a random variable.
A real number X connected with the outcome of a random experiment E. For example, if E consisits of two tosses the random variable which is the number of heads (0, 1 or 2).
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Outcome: |
HH |
HT |
TH |
TT |
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Value of X |
2 |
1 |
1 |
0 |
Ø Let S be the sample space associate with a given random experiment. A real- valued function defined on S and taking values in R is called a one – dimensional random variable.
Ø If the function values are ordered pairs of real numbers (vectors in two- space), the function is said to be a two dimensional random variable.
Ø More generally, an n-dimensional random variable is simply a function whose domain is S and whose range is a collection of n-tuples of real numbers (vectors in n-space).
Mathematical and rigorous definition of the random variable: Let us consider the probability space, the triplet (S, B, P), where S is the sample space, B is the σ-field of subsets in S, and P is a probability function on B.
A random variable is a function X(ω) with domain S and range such that for every real number a, the event .
Example 1: If a coin is tossed, then
, where
X(w) is a Bernoulli random variable. Here X(w) takes only two values.
Example 2: An experiment consists of rolling a die and reading the number of points on the upturned face. The most natural r.v. X to consider is
If we are interested in knowing whether the number of points is even or odd, we consider a random variable Y defined as follows:
Example 3: If a pair of fair dice is tossed then and n(S) = 36. Let X be a random variable with image set .
Remarks:
1. A function from S to R is a r.v. if and only if for real a, .
2. If X1 and X2 are random variables and C is a constant then CX1 , X1+X2, X1 .X2 are also random variables.
3. If X is a random variable then
(i) , where if ,
(ii)
(iii)
(iv) are random variables.
4. If X1 and X2 are random variables then and are also random variables.
5.If X is a r.v. and f(.) is a continuous function, then f(X) is a r.v.
6.If X is a r.v. and f(.) is a increasing function, then f(X) is a r.v.
7. If f is a function of bounded variations on every finite interval [a, b] and X is a r.v. then f(X) is a r.v.
Distribution Function : Let X be a r.v. The function F defined for all real x by
is called the distribution function (d.f) of the r.v. (X).
Remark: A distribution function is called the cumulative distribution function. The domain of the distribution function is and its range is [0, 1].
Properties of Distribution Function:
1. If F is the d.f. of the r.v. X and if a < b, then
(i)
(ii)
(iii)
2. If F is d.f. of one-dimensional r.v. X, then
(i) (ii)
3. If F is d.f. of one dimensional r.v. X , then
and
Discrete Random Variable:
A variable which can assume only a countable number of real values and for which the value which the variable takes depends on chance, is called a discrete random variable.
A real valued function defined on a discrete sample space is called a discrete random variable.
Example: marks obtained in a test,
number of accidents per month,
number of telephone calls per unit time, number of successes in n trials, and so on.
Probability Mass Function: If X is a one – dimensional discrete random variable taking at most a countably infinite number of values x1, x2, …… then its probabilistic behavior at each real point is described by a function called the probability mass function (or discrete density function).
Definition: If X is a discrete random variable with distinct values x1, x2, …… then the function p(x) defined as
is called the probability mass function of r.v. X.
Remarks: The number p(xi); I = 1, 2, … must satisfy the following conditions
(i)
(ii)
1. The set of values which X takes is called the spectrum of the random variable.
2. For discrete r.v., a knowledge of the probability mass function enables us to compute probabilities of arbitrary events.
Example 1: A random variable X has the following probability function:
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Values of X,x: |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
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p(x) |
0 |
k |
2k |
2k |
3k |
k2 |
2k2 |
7k2+k |
(i) Find k
(ii) Evaluate
(iii) If , find the minimum value of a,
(iv) Determine the distribution function of X.
Solution:
(i) Since
Since p(x) cannot be negative, k=-1 is rejected. Hence k = 1/10.
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Values of X,x: |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
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p(x) |
0 |
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(ii)
(iii)
(iv) The distribution function FX(x) of X is given by in the adjoin Table.
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Values of X,x: |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
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p(x) |
0 |
K |
3k |
5k |
8k |
8k+k2 |
8k+3k2 |
10k2+9k |
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FX(x) |
0 |
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Example 2: If, find (i) , and (ii)
Solution:
(i)
(ii)
Example 3: Two dice are rolled. Let X denote the random variable which counts the total number of points on the upturned faces, Construct a table giving the non-zero values of the probability mass function and draw the probability chart. Also find the distribution function of X.
Solution: If both dice are unbiased and the two rolls are independent, then each sample point of sample space S has probability 1/36. Then
These values are summarized in the following probability table:
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X: |
p(x) |
X |
p(x) |
|
2 |
1/36 |
8 |
5/36 |
|
3 |
2/36 |
9 |
4/36 |
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4 |
3/36 |
10 |
3/36 |
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5 |
4/36 |
11 |
2/36 |
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6 |
5/36 |
12 |
1/36 |
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7 |
6/36 |
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L7-Continuous Random Variable
A random variable X is said to be continuous if it can take all possible values (integral as well as fractional) between certain limits.
A random variable is said to be continuous when its different values cannot be put in 1-1 correspondence with a set of positive integers.
A continuous random variable is a random variable that can be measured to any desired degree of accuracy.
Example : age, height, weight, etc.
Probability Density Function: Consider the small interval (x, x+dx) of length dx round the point x. let f(x) be any continuous function of x so that f(x)dx represents the probability that X falls in the infinitesimal interval (x, x+dx).
Symbolically,
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p.d.f. fX(x) of the r.v. is defined as
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The probability for a variate value to lie in the interval dx is f(x)dx and hence the probability for a variate value to fall in the finite interval is :
Which represents area between the curve , x-axis and the ordinates at and .
The total probability is unity: , where [a, b] is the range of the random variable X.
The range of the variable may be finite or infinite.
Important Remark: (Difference between Discrete and Continuous Random Variable):
Various Measures of Central Tendency, Dispersion, Skewness and Kurtosis for Continuous Probability Distribution: Let fX(x) or f(x) be the p.d.f. of a r.v. X, where X is defined from a to b. then
(i)
(ii)
(iii)
(iv)
(v)
Example 1: The diameter of an electric cable, say X, is assumed to be a continuous random variable with p.d.f.:
(i) Check that f(x) is p.d.f. and
(ii) Determine a number b such that
Example 2: A continuous random variable X has a p.d.f.
Find a and b such that
(i) (ii)
Continuous Distribution Function:
If X is a continuous random variable with the p. d. f. f(x), then the function
is called the distribution function (d. f.) or sometimes the cumulative distribution function (c. f. d.) of the random variable X.
Properties of Distribution Function:
1.
2. F(x) is non-decreasing function of x.
3. F(x) is a continuous function of x on the right.
4. The discontinuities of F(x) are at the most countable.
5. It is denoted as
6. Similarly
Example: verify that following is a distribution function:
Example: The diameter, say X, of an electric cable, is assumed to be continuous random variable with p.d.f.
(i) Check that the above is a p.d.f.
(ii) Obtain an expression for the c. d. f. of X.,
(iii) Compute
(iv) Determine the number k such that .
