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Normal Distribution
Nagarajan Krishnamurthy
Introduction to Business Statistics for EPGP 2015-16 batchIndian Institute of Management Indore
Thanks to Prof. Arun Kumar and Prof. Ravindra Gokhale,co-instructors of QT1, AY 2012-13
Continuous Random Variable
Continuous random variable can take any value in an interval.
Probability Density Function
Probability distribution of a continuous random variable isdescribed by a probability density function.
Calculate Probability from Density Function
If X is a random variable with density function f (x) thenprobability that X will take values between a and b is thefollowing:
P(a ≤ X ≤ b) =
∫ b
a
f (x) dx
Properties of Probability Density Function
f (x) ≥ 0.
f (a) 6= P(X = a).
Probability of a Continuous Random Variable at a
Point
Probability that a continuous random variable takes aparticular value is zero.
Example
A random variable X has a probability density function
f (x) = x2, 0 ≤ x ≤ 1
Is it a legitimate probability density function?
Example Continued
If f (x) = cx2, 0 ≤ x ≤ 1. Find a value of c so that f (x) is alegitimate probability density function.
Mean of Continuous Random Variable, also known
as the first moment
µ =
∫ b
a
x f (x) dx ,
where
f (x) is the density function.
[a, b] is the domain of the random variable.
Higher Moments of Continuous Random Variable
E (X n) =
∫ b
a
xn f (x) dx ,
where
f (x) is the density function.
[a, b] is the domain of the random variable.
n is an integer.
Variance of Continuous Random Variable
σ2 =
∫ b
a
(x − µ)2f (x) dx
, where
f (x) is the density function.
[a, b] is the domain of the random variable.
Normal Probability Distribution
Normal distribution is very important because
Outcomes of a large number of real life situations has abell shaped frequency distribution that can be modeled bya normal distribution.
Central Limit Theorem.
Normal Probability Density Function
f (x) =1
σ√2π
e−(x−µ)2
2σ2 ,−∞ < x <∞
Notation for a Normal Random Variable
N(µ, σ2)
Mean and Variance of Normal Random Variable
Mean = µ
Variance = σ2.
Probability Calculation for a Normal Random
Variable
P(a ≤ x ≤ b) =
∫ b
a
1
σ√2π
e−(x−µ)2
2σ2 dx
Warning: Don’t try to calculate this integral.
Standard Normal Random Variable
Normal random variable with mean 0 and standard deviation 1.
Notation: Z
Finding Probability for Standard Normal Random
Variable Using Z-table
Exercise 1
Calculate the area under the standard normal curve to the leftof these values:a) z = 1.6
Exercise 1
b) z = 1.83
c) z = 0.90
Exercise 1
d) z = 4.18
Finding Probability for any Normal Random
Variable Using Z-Table
Let’s say X has a normal distribution with mean µ andstandard deviation σ then P(X ≤ a) = P(Z ≤ a−µ
σ).
Exercise 2
A normal random variable X has mean µ = 1.20 and standarddeviation σ = 0.15. Find the following probabilities.a) P(X < 1.10)
Exercise 2
b) P(X > 1.38)
Exercise 2
c) P(1.35 < X < 1.50)
Finding Percentile of Standard Normal Random
Variable
Exercise 3
a)Find a z0 such that P(Z > z0) = 0.025.
Exercise 3
b)Find a z0 such that P(Z < z0) = 0.9251.
Exercise 3
Find a z0 such that P(−z0 < Z < z0) = 0.8262.
Finding Percentile of any Normal Random Variable
Exercise 4
A normal random variable X has mean 35 and standarddeviation 10. Find a value X that has area 0.01 to its right.This is the 99th percentile of this normal distribution.