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What is a Random Variable?
Types of Random Variable
Discrete Random Variable
Continuous Random Variable
Probability Mass Function
What is Probability Distribution?
Cumulative Distribution Function (C.D.F) of Discrete Random Variable
Problems
Probability Density Function
Distribution Function
A Random Variable or Stochastic Variable or Random quantity in the field of probability and statistics is a variable quantity, whose possible values depend on a set of random outcome events in random manner.
The outcome space is defined in terms of set theory or a set.
A random variable is a function associating a real number with each outcome of a sample space S of a random experiment and it is denoted by X.
Domain of a random variable X is a sample space S and codomain or range of a random variable X is the set of real values taken by X, Thus, symbolically X: S-> R.
There are two types of Random Variable:
A random variable X which can assume countable number of isolated values (integers) is called discrete random variable. The meaning of the word ‘discrete’ is separate and individual
Examples:
Number of kids in a family
Number of attempts to hit a target
Number of heads in tossing a coin thrice
Number of stars in the sky
A random variable X which can assume all real values within a given interval is known as continuous random variable. Thus, the possible values of a continuous random variable are uncountably infinite.
Price of a commodity
Age of a person
Height of a person in cm
Life of an electric component (in hours)
Examples to identify random variables as either discrete or continuous in each of the following situations:
Q1. A page in a book can have at most 300 words
X= Number of misprints on a page
Solution
X = Number of misprints on a page
Since a page in a book has at most 300 words, X takes the finite values
Therefore, random variable X is discrete
Range = {0, 1, 2….299, 300}
Q2. A gymnast goes to the gymnasium regularly
X = Reduction of his weight in a month
X = Reduction of weight in a month
X takes uncountable infinite values
Therefore, random variable X is continuous.
If X is the discrete random variable taking values x_{1}, x_{2}, x_{3}…x_{n }and we find the values of P(X = x_{1}), P(X = x_{2}), P(X = x_{3})…P(X = x_{n}), then we obtain the functions P(x) = P(X = x) for x = x_{1}, x_{2}, x_{3}…x_{n}.
This function is called Probability Mass Function (p.m.f) of the random variable X.
Example
If two coins are tossed and X=number of heads, then the sample space of tossing two coins is {HH, HT, TH, TT}
Therefore, P(X = 0) = P (0) =1/4, P(X = 1) = P(1) = 2/4 = 1/2
P(X = 2) = P (2) = 1/4
Hence, the probability mass function is given by
P(x) = 1/4, x = 0
=1/2, x = 1
= ¼, x = 2
For the sample space S = {0, 1, 2} for X.
Note:
If X is any random variable taking values from sample space S and P(x) is the probability mass function of X such that x ∈ S, then
0 ≤ P(x) ≤ 1 for all x ∈ S
∑ P(x) = 1, that is, the total of all values of P(x) for x ∈ S is always unity.
We can verify this in the example given above
∑ P(x) = P(0) + P(1) + P(2) = 1/4 + 1/2 + ¼ =1
If X is a discrete random variable taking values x_{1}, x_{2}, x_{3}….x_{n} with corresponding probabilities p_{1}, p_{2}, p_{3}…p_{n}, then the set of ordered pairs (x_{i}, p_{i}), i = 1, 2, 3….n is called a Probability Distribution of the random variable X.
If the two coins are tossed and X is the number of heads, then the probability distribution of the random variable X can be given as below:
Note that the sum of all probabilities
= ¼ + 2/4 + ¼ = 4/4 = 1
In general case, the probability distribution of discrete random variable X can be given as:
Where ∑ p_{i }= p_{1 }+ p_{2 }+ p_{3 }+….+ p_{n }= 1
Let X be a discrete random variable and its probability distribution is as follows:
The Cumulative Distribution Function (c.d.f) of X is denoted by F(x) and it is defined as
F(x) = P[X ≤ x], x ∈ R
Notes:
Domain of c.d.f is R
Codomain of c.d.f is [0, 1]
F(x_{i}) = P[X ≤ x_{i}] = p_{1 }+ p_{2 }+ p_{3 }+….p_{i}, i = 1, 2, 3,…n
c.d.f of a discrete random variable X is also represented in tabular form as follows:
Q1. Three balanced coins are tossed simultaneously. If X denotes the number of heads, find probability distribution of X.
Sol. When three balanced coins are tossed then the sample space is
{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
X denotes the number of heads.
X can take the values 0, 1, 2, 3
P[X = 0] = P (0) = 1/8 P [X = 1] = P (1) = 3/8
P[X = 2] = P (2) = 3/8 P [X = 3] = P (3) = 1/8
Q2. Given below is the probability distribution of X:
Find the value of k
P (X ≥ 2), P(X < 3 ), P(X≤1)
Obtain the c.d.f of X
Solution:
∑ P[X = x] =1
P[X = 0] + P[X = 1] + P[X = 2] + P[X = 3] + P[X = 4] = 1
k + 2k + 4k + 2k + k =1
10k = 1
k = 1/10
= 4k + 2k + k = 7k
= 7 (1/10) = 7/10
P (X<3) = P[X=0] +P [X=1] + P[X=2]
= k + 2k+ 4k = 7k
= 7 (1/ 10) = 7/10
P (X≤ 1) = P[X=0] + P [X=1]
= k +2k = 3k
= 3 (1/10) = 3/10
F(x) = P (X ≤ x)
F(0) = P (X ≤ 0) = P(0) = k = 1/10
F(1) = P(X ≤ 1) = P(0) + P(1) = k + 2k = 3k = 3/10
F(2) = P(X ≤ 2) = P(0) + P(1) + P(2) = k + 2k + 4k = 7k = 7/10
F(3) = P(X ≤ 3) = P(0) + P(1) + P(2) + P(3) = k + 2k + 4k + 2k = 9/10
F(4) = P(X ≤ 4) = P(0) + P(1) + P(2) + P(3) + P(4) = k + 2k + 4k + 2k + k = 10/10 = 1
Hence, the c.d.f of X is as follows:
Probability Density Function (P. D. F) and Distribution Function (D. F) of a Continuous Random Variable
A real valued function f(x) is called a Probability Density Function (P. D. F) of a continuous random variable X, if it satisfies the following:
F(x) ≥ 0 for all x ∈ R
If X takes the values in the interval (a,b) then the function f(x) is such that
F(x) ≥ 0 for a < x < b, and
∫^{b}_{a} f(x) dx =1
The area under the curve y= f(x) bounded by the X-axis and the coordinates x = a and x = b is 1, because it represents the total probability P(a< X <b) which is equal to 1.
Also, P (p < X < q) is area under the curve y = f(x) bounded by the X-axis and the coordinates x = p and x = q which is shaded in the figure.
The graph of the P. D. F of R. V. X is called the Probability Curve or Probability Density Curve.
Assume X as a continuous random variable with probability density function f(x), then the cumulative distribution function F(x)of X is defined for every real number x_{i}, as F(x_{i}) =P [X ≤ x_{i}] = ∫ f(x) dx
Remarks:
F(x_{i}) is the area under the curve y= f(x) to the left of x as shown in the figure right:
F(x) increases smoothly as x increases
If range of X is (a,b) then F(x) = 0, for x< a and F(x) = 1, for x≥b
P [X >x]=1 - P [X ≤ x_{i}]= 1- F(x)
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