One transformed variable If X N(mu,sigma2) and Y=aX+b, then Y N(amu+b,a2sigma2). Independent normal variables If X N(_X, _X2) and Y N(_Y, _Y2) are independent, then aX+bY N(a _X+b _Y, a2 _X2+b2 _Y2). **Quantity** | **Distribution** S_n=X_1+. s+X_n | S_n N(nmu,nsigma2) X=(X_1+. s+X_n)/(n) | X N(mu,(sigma2)/(n)) These exact results require X_1,,X_n to be independent observations from N(mu,sigma2). Independence is also what removes covariance terms when variances are added. Translate the context before calculating - The total mass of three independently chosen durians is X_1+X_2+X_3. - Three times the mass of one chosen durian is 3X; this scales one observation. - “The total mass of four pumpkins exceeds four times one durian mass” becomes Y_1+Y_2+Y_3+Y_4>4X. An affine transformation of a normal variable is normal. An affine transformation aB+b of B (n,p) is not generally binomial.

If \(X\sim N(\mu,\sigma^2)\) and \(Y=aX+b\), then \(Y\sim N(a\mu+b,a^2\sigma^2)\).

Affine transformation

If \(X\sim N(\mu_X,\sigma_X^2)\) and \(Y\sim N(\mu_Y,\sigma_Y^2)\) are independent, then \[aX+bY\sim N(a\mu_X+b\mu_Y,\ a^2\sigma_X^2+b^2\sigma_Y^2).\]

Linear combination

These exact results require \(X_1,\ldots,X_n\) to be independent observations from \(N(\mu,\sigma^2)\). Independence is also what removes covariance terms when variances are added.

The total mass of three independently chosen durians is \(X_1+X_2+X_3\).

Three times the mass of one chosen durian is \(3X\); this scales one observation.

“The total mass of four pumpkins exceeds four times one durian mass” becomes \(Y_1+Y_2+Y_3+Y_4>4X\).

An affine transformation of a normal variable is normal. An affine transformation \(aB+b\) of \(B\sim\operatorname{Bin}(n,p)\) is not generally binomial.

Quantity	Distribution
\(S_n=X_1+\cdots+X_n\)	\(S_n\sim N(n\mu,n\sigma^2)\)
\(\displaystyle\overline X=\frac{X_1+\cdots+X_n}{n}\)	\(\displaystyle\overline X\sim N\left(\mu,\frac{\sigma^2}{n}\right)\)

Linear Transformations and Combinations

Linear Transformations and Combinations