]]>
LearnNext
Get a free home demo of LearnNext

Available for CBSE, ICSE and State Board syllabus.
Call our LearnNext Expert on 1800 419 1234 (tollfree)
OR submit details below for a call back

clear

Variance of a Random Variable

2,413 Views
Have a doubt? Clear it now.
live_help Have a doubt, Ask our Expert Ask Now
format_list_bulleted Take this Lesson Test Start Test

Variance of a Random Variable - Lesson Summary

In statistics, variance is a measure of the spread or scatter in data.

Variance can also be a measure of the spread of the values of a random variable in a probability distribution.

Var(X) = σx2 = ∑i=1n(xi - μ)2p(xi)

x1,x2,x3,.....,xn are the possible values of X and they occur with the probabilities p(x1), p(x2),p(x3), ...p(xn), respectively.

μ = E(X) = Mean of X

σx2 = E(X - μ)2

The square root of the variance of random variable X is called the standard deviation of X.

σx = √Var(X) = √(∑i=1n(xi - μ)2p(xi))

Var(X) = σx2 = ∑i=1n(xi - μ)2p(xi)

= ∑i=1n(xi2 + μ2 - 2μxi)p(xi)

= ∑i=1n xi2 p(xi) + ∑i=1n μ2 p(xi) - ∑i=1n 2μxi p(xi)


= ∑i=1n xi2 p(xi) + ∑i=1n μ2 p(xi) - 2μ ∑i=1n xi p(xi)

= ∑i=1n xi2 p(xi) + μ2 - 2μ2 ['.' ∑i=1n p(xi) = 1 and μ = ∑i=1n xi p(xi) ]

= ∑i=1n xi2 p(xi) - μ2

= ∑i=1n xi2 p(xi) - [∑i=1n xi p(xi)]2

⇒ Var(X) = E(X2) - [E(X)]2

Ex:

The probability distributions of two experiments are

  X

 0

  1

  2

  3

   Y

  ‑ 1

  0

  3

  5

  6

 P(X)

1/8

3/8

3/8

1/8

  P(Y)

 2/8

 3/8

1/8

1/8

1/8

Calculate the mean of the random variables in both the cases.

Mean of X = E(X) = ∑i=14 xi pi

= x1 p1 + x2 p2 + x3 p3 + x4 p4

= (0 x 1/8) + (1 x 3/8) + (2 x 3/8) + (3 x 1/8)

= 0 + 3/8 + 6/8 + 3/8

= (3+6+3)/8

= 12/8 = 1.5

Mean of Y = E(Y) = ∑i=15 xi pi

= x1 p1 + x2 p2 + x3 p3 + x4 p4 + x5 p5

= (-1 x 2/8) + (0 x 3/8) + (3 x 1/8) + (5 x 1/8) + (6 x 1/8)

= -2/8 + 0 + 3/8 + 5/8 + 6/8

= (-2+3+5+6)/8

= 12/8 = 1.5

Observe that both the distributions have the same mean.

It does not give us any information about the variability in the values of the random variables.

To understand the variability in the variables, draw two graphs using the distributions ... by taking the values of the random variables along the X-axis and their probability along the Y-axis.

Var(X) = σx2 = ∑i=14(xi - μ)2p(xi)

= (0-1.5)2 x 1/8 + (1 - 1.5)2 x 3/8 + (2 - 1.5)2 x 3/8 + (3 - 1.5)2x 1/8

= (-1.5)2 x 1/8 + (- 0.5)2 x 3/8 + (0.5)2 x 3/8 + (1.5)2x 1/8

= 2.25 x 1/8 + 0.25 x 3/8 + 0.25 x 3/8 + 2.25 x 1/8

= 0.281 + 0.094 + 0.094 + 0.281

= 0.469

Var(Y) = σy2 = ∑i=14(yi - μ)2p(yi)

= (-1 -1.5)2 x 2/8 + (0 - 1.5)2 x 3/8 + (3 - 1.5)2 x 1/8 + (5 - 1.5)2x 1/8 + (6 - 1.5)2x 1/8

= (-2.5)2 x 2/8 + (- 1.5)2 x 3/8 + (1.5)2 x 1/8 + (3.5)2x 1/8 + (4.5)2x 1/8

= 6.25 x 2/8 + 2.25 x 3/8 + 2.25 x 1/8 + 12.25 x 1/8 + 20.25 x 1/8

= 1.562 + 0.844 + 0.781 + 2.531 + 3.781

= 6.749

⇒ Var(Y) > Var(X)

The spread of values is more in Y than in X.

Comments(0)

Feel the LearnNext Experience on App

Download app, watch sample animated video lessons and get a free trial.

Desktop Download Now
Tablet
Mobile
Try LearnNext at home

Get a free home demo. Book an appointment now!

GET DEMO AT HOME