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Question
Given that `barx` 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 `abarx` and a2 σ2, respectively (a ≠ 0).
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Solution
Here `barx` = `(x_1 + x_2 + x_3 + ... + x_n)/n = (sumx)/n`
Also, `(x_1^2 + x_2^2 + x_3^2 + ... + x_n^2)/n = (sumx^2)/n`
New mean = `(ax_1 + ax_2 + ax_3 + ... + ax_n)/n`
= `a((x_1 + x_2 + x_3 + ... + x_n))/n = abarx`
Also,
σ2 = `(n(x_1^2 + x_2^2 + ... + x_n^2) - (x_1 + x_2 + ... + x_n)^2)/n^2`
∴ New variance
`(n(a^2x_1^2 + a^2x_2^2 + a^2x_3^2 + ... + a^2x_n^2) - (ax_1 + ax_2 + ax_3 + ... + ax_n)^2)/n^2`
= `a^2 [[n(x_1^2 + x_2^2 + x_3^2 + ... + x_n^2) - (x_1 + x_2 + + x_3 ... + x_n)^2)/n^2]`
= a2σ2
Hence proved.
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