Question

Let X be a random variable and let EX = ? show that (a) E(X - ?)^2 = E(X^2) - ?^2. (b) Var (aX + b) = a^2 Var (X) If X ~ N(?, ?^2), show (X - ?) / ? ~ N(0, 1).

          Let X be a random variable and let EX = ? show that
(a) E(X - ?)^2 = E(X^2) - ?^2.
(b) Var (aX + b) = a^2 Var (X)
If X ~ N(?, ?^2), show
(X - ?) / ? ~ N(0, 1).

Added by Lori B.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Aman Gupta

09/21/2023

Step 1

We know that $E[(X-p)^2] = E[X^2 - 2pX + p^2]$. Using the linearity of expectation, we have: $E[X^2 - 2pX + p^2] = E[X^2] - 2pE[X] + p^2$. Since $E[X] = p$, we can substitute this back into the equation: $E[X^2] - 2p^2 + p^2 = E[X^2] - p^2$. So, we have Show more…

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Let X be a random variable and let EX = μ show that (a) E(X - μ)^2 = E(X^2) - μ^2. (b) Var (aX + b) = a^2 Var (X) If X ~ N(μ, σ^2), show (X - μ) / σ ~ N(0,1).