3. Types of Random Distributions
Discrete Random Distribution
Continuous Random Distribution
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4. Probability Distribution of
Random Variable
Probability Distribution of Random Variable is
defined as a table that depicts all the possible
values of random variable along with their
probabilities. Probability distribution of a
discrete random X can be expressed as follows:
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X …….. Total
P(X) ……..
5. Types of Probability Distribution
Discrete Random Distribution:
A random variable is said to be discrete if it takes only a finite
or an infinite but countable number of values.
Probability Function of a Discrete Random Distribution: If
for a random variable X, the real valued function p(x) is such
that
P(X=x) = p(x),
Then p(x) is called probability function or probability density
function of a random discrete variable. Probability function
p(x) gives the measure of probability for different values of
X.
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6. Properties of a Probability Function
If p(x) is a probability function of a random variable
X, then it possesses the following properties:
1. p(x) is positive for all values of x i.e. p(x) ≥ 0 for
all x.
2. ∑ p(x) = 1, summation is taken over for all values
of x.
3. p(x) measures the probability for any given value
of x.
4. p(x) can not be negative for any value of x.
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8. Continuous Random Variable
A continuous random variable is a random
variable that can take on any value in an interval
of two values.
Height, weight, length etc. are some of the
examples.
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9. Probability Density Function of a Continuous Random Variable
The probabilities associated with a continuous
random variable X are determined by the
probability Distribution function f(x) of random
variable X. where
1. f(x) ≥ 0 for all values of x
2. The probability that x will lie between two numbers a
and b is equal to the area under the curve y = f(x)
between x=a and x=b
3. The total area under the entire curve y = f(x) is always
equal to unity i.e. 1.
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10. Cumulative Distribution Function of a Continuous Random Variable
Cumulative Distribution Function of a Continuous
Random Variable is F(x), where
F(x) = P(X=x)=Area under the curve y= f(x) between
the smallest value of X ( often -∞ ) and a point x.
Properties of Cumulative Distribution Function:
1. The CDF F(X) is smooth.
2. It is a non-decreasing function that increases from 0
to 1.
3. Expected value or mean is denoted by E(X)
4. The variance is denoted by V(X)
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12. Theorems on Mathematical
Expectation
Theorem 1: Expected value of a constant is
constant, that is if C is constant, then
E(C) = C
Theorem 2: If C is constant, then
E(CX) = C. E(X)
Theorem 3: If a and b are constants, then
E(a X ± b) = a . E(X) ± b
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14. Variance
Variance of the probability distribution of a
random variable X is the mathematical expectation
of 𝑋 − 𝐸 𝑋 2 . Then
Var(X) = E 𝑋 − 𝐸 𝑋 2
If we put E(X) = μ then Var(X) = E 𝑋 − μ 2
Var(X) = E(𝑋2
) - μ2
For Standard Deviation ( 𝜎) just find out the
square root of the equations.
15. Theorems on Variance
Theorem 1: If C is constant, then
V(CX) = C2
V(X)
Theorem 2: If C is constant, then
V(C) = 0
Theorem 3: If a and b are constants, then
V(a X + b) = a2
. V(X)
Theorem 4: If X and Y are two independent
random variables, then
(i) V(X+Y)= V(X) + V(Y)
(ii) V(X-Y)= V(X) + V(Y)
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16. Examples
1. The probability function of a random variable X is p(x) =
2𝑥+1
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, x = 1, 2, 3, 4, 5, 6. Verify whether p(x) is a probability
function?
2. For a random variable X, p(x) =
𝑥
𝑥+1
, where x = 1, 2, 3. Is
p(x) a probability density function.
3. The probability distribution of a random variable x is given
below. Find (1) E(x), (2) V(x), (3) E (2x-3) and (4) V(2x-3)
(A- 0, 1.6, -3, 6.4)
4. Amit plays a game of tossing a die. If the number is less
than 3 appears, he is getting Rs. A, otherwise he pays Rs.
10. If the game is fair, find a. (a=20)16
x -2 -1 0 1 2
p(x) 0.2 0.1 0.3 0.3 0.1
