Probability Distribution Mean (Expectation), Variance :: Problems

Problem Back to Problems Page
 
If the variance of a random variable x is 5, then the variance of the random variable
(− 3x) is

Net Answers :
[Var(-3x): 45]

Solution  
 

We know, var (x) = E(x2) − (E (x))2
= Σ px2 − (Σ px)2

Therefore,

var (x) = 5
⇒ Σ px2 − (Σ px)2 = 5 → (1)
var ( − 3x) = Σ p (− 3x)2 − {Σ p (− 3x)}2
= Σ p {(− 3)2 × (x)2} − {Σ − 3px}2
= Σ p {9x2} − {Σ − 3px}2
= Σ 9px2 − {− 3 Σ px}2
= 9 × Σ px2 − {(− 3)2 × (Σ px)2}
= 9 Σ px2 − {9 (Σ px)2}
= 9 [ Σ px2 − (Σ px)2]
= 9 [5]       [Using (1)]
= 45

Alternative

We know, var (x) = E(x2) − (E (x))2
= Σ px2 − (Σ px)2
var (ax) = a2 var (x)
Given,

var (x) = 5 → (A)
Therefore,

var (− 3x) = (− 3)2 var (x)
= 9 var (x)
= 9 × 5       [Using (A)]
= 45

Credit : Vijayalakshmi Desu

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