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.