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Chapter 7: Q.7.31 (page 366)

For random variables X and Y, show that

Var鈦(X+Y)鈮Var鈦(X)+Var鈦(Y)

That is, show that the standard deviation of a sum is always less than or equal to the sum of the standard deviations.

Short Answer

Expert verified

The standard deviation is the square root of variance that isVar鈦(X+Y)鈮Var鈦(X)+Var鈦(Y)

Step by step solution

01

Given information

X and Y are random variable

That has the equationVar鈦(X+Y)鈮Var鈦(X)+Var鈦(Y)

02

Solution 

We need to define the following equations first,

Var鈦(X)=蟽x2

Var鈦(Y)=蟽y2

The relation of variance and covariance show that,

Var鈦(X+Y)=蟽x2+蟽y2+2Cov鈦(X,Y)

Var鈦(X+Y)=蟽x2+蟽y2+2蟽x蟽y

The property of correlation Shows that,

Corr鈦(X,Y)=Cov鈦(X,Y)蟽x蟽y

The preceding inequality will becomes,

Corr鈦(X,Y)=Cov鈦(X,Y)蟽x蟽y鈮1

So,

Cov鈦(X,Y)蟽x蟽y鈮1

Cov鈦(X,Y)鈮蟽x蟽y

03

Solution

Now use the above results to obtain the result,

Var鈦(X+Y)=蟽x2+蟽y2+2Cov鈦(X,Y)

Var鈦(X+Y)鈮蟽x2+蟽y2+2蟽x蟽y

Var鈦(X+Y)鈮蟽x+蟽y2

Var鈦(X+Y)鈮(Var鈦(X)+Var鈦(Y))2

04

Final answer

The standard deviation is the square root of variance that isVar鈦(X+Y)鈮Var鈦(X)+Var鈦(Y)

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