0
You visited us 0 times! Enjoying our articles? Unlock Full Access!
Question

Given that ¯X is the mean and σ2 is the variance of n observations X1,X2...Xn. Prove that the mean and variance of the observations aX1,aX2,aX3....aXn are ¯ax and a2σ2 respectively (a0).

Solution
Verified by Toppr

The given n observation are x1,x2..xn
Mean = ¯x
variance = σ2
σ2=1nni=lyi(xi¯x)2 ....(i)
If each observation is multiplied by a and the new observation are yi then
yi=axi,i.e,xi=1ayi
¯y=1nni=lyi=1nni=laxi=anni=lxi=¯ax,(¯x=1nni=1xi)
Therefore mean of the observation, ax1,ax2....axn is ¯ax
Substituting the values of xi and ¯x in (1) we obtain
σ2=12ni=l(1ayi1a¯y)2
a2σ2=1nni=l(yi¯y)2
Thus the variance of the observation ax1,ax2...axn is a2σ2

Was this answer helpful?
9
Similar Questions
Q1
Given that ¯X is the mean and σ2 is the variance of n observations X1,X2...Xn. Prove that the mean and variance of the observations aX1,aX2,aX3....aXn are ¯ax and a2σ2 respectively (a0).
View Solution
Q2

Given that (¯¯¯x) is the mean and σ2 is the variance of n observations x1,x2,....xn. Prove that the mean and variance of the observations ax1,ax2,ax3...axnare a¯¯¯x and a2σ2 respectively (a0)

View Solution
Q3

If the mean and variance of the observations x1, x2, x3, , xn are ¯x and σ2 respectively and a be a nonzero real number, then show that the mean and variance of ax1, ax2, ax3, . axn are ¯ax and a2 σ2 respectively.

View Solution
Q4
Given that is the mean and σ 2 is the variance of n observations x 1 , x 2 … x n . Prove that the mean and variance of the observations ax 1 , ax 2 , ax 3 … ax n are and a 2 σ 2 , respectively ( a ≠ 0).
View Solution
Q5

Let x1, x2, ,xn be n observations, and let ¯x be their arithmetic mean and σ2 be the variance.
Statement – 1: Variance of 2x1,2x2,,2xn is 4σ2
Statement – 2: Arithmetic mean 2x1,2x2,,2xn is 4¯x


View Solution