Random Variable

Random Variable
Bipul Kumar Sarker
Lecturer
BBA Professional
Habibullah Bahar University College
Chapter-02

Definitions and basic concepts:
Bipul Kumar Sarker, Lecturer (BBA Professional), HBUC
The following definitions and terms are used in studying the theory of probability distribution.
Random variable:
A variable whose value is a number determined by the outcome of a random
experiment is called a random variable.
There are two types of random variable in statistics such as follow:
 Discrete Random Variable
 Continuous Random Variable

Discrete random variable:
Discrete random variable:
If a random variable takes only a finite or a countable number of values, it is called a
discrete random variable.
Example:
When 3 coins are tossed, the number of heads obtained is the random variable X
assumes the values 0,1,2,3 which form a countable set. Such a variable is a discrete
random variable.

Continuous random variable:
A random variable X which can take any value between certain interval is called a
continuous random variable.
Example:
The height of students in a particular class lies between 4 feet to 6 feet.
We write this as X = {x|4 x 6}
The maximum life of electric bulbs is 2000 hours. For this the continuous random
variable will be X = {x | 0 x 2000}

2.2 Probability mass function:
Let X be a discrete random variable which assumes the values 𝑥1, 𝑥2, … … … 𝑥 𝑛 with each of
these values, we associate a number called the probability 𝑃𝑖 = 𝑃(𝑋 = 𝑥1), i = 1,2,3…n.
This is called probability of 𝑥𝑖 satisfying the following conditions.
i. Pi ≥ 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖, 𝑖. 𝑒 𝑃𝑖
′
𝑠 𝑎𝑟𝑒 𝑎𝑙𝑙 𝑛𝑜𝑛 − 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒.
ii. 𝑃𝑖 = 𝑃1 + 𝑃2 + ⋯ … … … … … … . . +𝑃𝑛 = 1 𝑖. 𝑒 𝑡𝑕𝑒 𝑡𝑜𝑡𝑎𝑙 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑖𝑠 𝑜𝑛𝑒
This function Pi or 𝑃(𝑋𝑖) is called the probability mass function of the discrete
random variable X.
The set of all possible ordered pairs (x, p(x)) is called
the probability distribution of the random variable X.

Note:
The concept of probability distribution is similar to that of frequency distribution. Just as
frequency distribution tells us how the total frequency is distributed among different
values (or classes) of the variable, a probability distribution tells us how total probability
1 is distributed among the various values which the random variable can take. It is usually
represented in a tabular form given below:
X 𝑥1 𝑥2 𝑥3 ………….. 𝑥 𝑛
P(X = x) 𝑃(𝑥1) 𝑃(𝑥2) 𝑃(𝑥3) ………….. 𝑃(𝑥 𝑛)

2.2.1 Discrete probability distribution:
If a random variable is discrete in general, its distribution will also be discrete.
For a discrete random variable X, the distribution function or cumulative distribution is
given by F(x) and is written as
F(x) = P(X x) ; - < x < 
Thus in a discrete distribution function, there are a countable number of points
𝑥1, 𝑥2, … … … and their probabilities Pi such that
𝐹 𝑥𝑖 = Pi
𝑥 𝑖< 𝑥
; 𝑖 = 1,2,3, … … … … . 𝑛

2.2.2 Probability density function (pdf):
A function f is said to be the probability density function of a continuous random
variable X, if it satisfies the following properties.
i. 𝒇 𝒙 0 ; -< x < 
ii. 𝒇 𝒙 = 𝒇 𝒙 𝒅𝒙 = 𝟏
∞
−∞

Remark:
In case of a discrete random variable, the probability at a point ie P(x = a) is not zero
for some fixed ‘ a’ However in case of continuous random variables the probability at a
point is always zero i.e
𝑷 𝑿 = 𝒂 = 𝒇 𝒙 𝒅𝒙 = 𝟎
𝒂
𝒂
Hence P( a X b) = P(a < X < b) = P(a X < b) = P(a < X b)
The probability that x lies in the interval (a,b) is given by
𝑷 𝒂 < 𝑿 < 𝒃 = 𝒇(𝒙)
𝒃
𝒂
𝒅𝒙

Distribution function for continuous random variable:
If X is a continuous random variable with p.d.f f(x), then the distribution function
is given by
i. 𝑭 𝑿 = 𝒇 𝒙
𝒙
−∞
𝒅𝒙 = 𝑷 𝑿 ≤ 𝒙 ; −∞ < 𝐗 < ∞
i. 𝑭 𝒃 − 𝑭 𝒂 = 𝒇 𝒙
𝒃
𝒂
𝒅𝒙 = 𝑷 𝒂 < 𝑿 < 𝒃

Definitions and basic concepts:

Random Variable

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