[Math] Deriving Mean and Variance of (constant * Gaussian Random Variable) and (constant + Gaussian Random Variable)

integrationnormal distributionprobability

If X is a Gaussian, N($\mu_x, \sigma_x^2$), then I came across the following properties:

1) Y = X + k, k is a constant. Then Y is normally distributed as N($\mu_y, \sigma_y^2$) such that:

$\mu_y = \mu_x + k$

$\sigma_y = \sigma_x$

2) Y = kX, k is a constant. hen Y is normally distributed as N($\mu_y, \sigma_y^2$) such that:

$\mu_y = k*\mu_x$

$\sigma_y = |k|*\sigma_x$

Is there a proof for the above two statements? Thanks in advance.

{
Edit: I have modified the original question.

Original question: Deriving Mean and Variance of (constant * Gaussian Distribution).

Due to some misunderstandings, I came to know that the above is not a feasible PDF and that what I actually want is: Deriving Mean and Variance of (constant * Gaussian Random Variable) and (constant + Gaussian Random Variable).
}

Edit 2:

If X is Gaussian, prove kX is Gaussian where k is a constant such that k<0.

$P(kX < x)$

$ = P(X > \frac{x}{k})$ (since k<0)

$ = \int_{\frac xk}^\infty \frac{1}{\sqrt{2\pi}\sigma}\exp \left(-\frac{(t-\mu)^2}{2\sigma^2}\right)dt$

I am not able to resolve the limits to get the resulting equation into the form for Cumulative Density Function. How do I proceed?

Best Answer

The expressions for the mean and standard deviation are true for any random variable (EDIT: for which $EX$ and $VX$ are defined): $$ E(X+k) = EX+Ek = EX + k, \quad V(X+k)=VX+Vk=VX, $$ $$ E(kX) = kEX, \quad V(kX) = k^2VX $$ I am sure you can find proofs for these easily. But we still need to show that the transformed variables have normal distributions. We can do that as follows. Assume $k>0$. Then: $$ P(kX<x) = P(X<\frac xk) = \int_{-\infty}^\frac xk \frac{1}{\sqrt{2\pi}\sigma}\exp \left(-\frac{(t-\mu)^2}{2\sigma^2}\right)dt $$ $$ = \int_{-\infty}^x \frac{1}{\sqrt{2\pi}k\sigma}\exp \left(-\frac{(u -k\mu)^2}{2(k\sigma)^2}\right)du $$ where we substituted $u=kt$. This is the CDF of $\mathcal N(k\mu, k^2\sigma^2)$, so $kX$ has that distribution. (Note that we rediscovered the mean and variance). The cases $kX$ for $k<0$, and $X+k$ can be shown similarly.

EDIT: The case of $kX$, $k<0$. You showed $$ P(kX<x) = \int_{\frac xk}^{\infty} \frac{1}{\sqrt{2\pi}\sigma}\exp \left(-\frac{(t-\mu)^2}{2\sigma^2}\right)dt $$ We substitute $u=kt$, so $du=kdt$. The lower limit becomes $k\frac xk = x$, and the upper limit becomes $k\infty = -\infty$ (you can rephrase this as a limit, if you want). Plugging in: $$ \int_x^{-\infty} \frac 1k \cdot \frac{1}{\sqrt{2\pi}\sigma}\exp \left(-\frac{(\frac uk -\mu)^2}{2\sigma^2}\right)du $$ $$ = -\int_x^{-\infty} \frac 1{|k|} \cdot \frac{1}{\sqrt{2\pi}\sigma}\exp \left(-\frac{(u -k\mu)^2}{2k^2\sigma^2}\right)du $$ $$ = \int_{-\infty}^x \frac{1}{\sqrt{2\pi}|k|\sigma}\exp \left(-\frac{(u - k\mu)^2}{2(|k|\sigma)^2}\right)du $$ which is the CDF for $\mathcal N(k\mu,(|k|\sigma)^2)$.

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