Probability Distributions "related" to normal distributions

Question

I am tutoring a student and I am unable to determine what "related" means in this problem. Im thinking just the Z and students t are normally distributed but arent chi squared and the F test related to a normal probability distribution even though they are not normally distributed? I read a scientific paper on the G distribution and still cant tell if it is normally distributed or "related" to a normal distribution. Neither can i make sure about morphets q. Can anyone enlighten me please.

Choose all probability distributions which are theoretically related to the normal probability distribution.

A) G

B) Morphet's Q

C) Chi Square

D) Z

E) Student's t

F) F

I am claiming that only Z and Students t are normally distributed

Answer 1

Student's t, Chi-squared, and F are 'related' to normal, even though they are not normally distributed.

Let $Z_0, Z_1, \dots, Z_n$ be independently standard normal. Then $Q = \sum_{i=1}^n Z_i^2 \sim \mathsf{Chisq}(n),$ the chi-squared distribution with $n$ degrees of freedom.

Also, $T = \frac{Z_0}{\sqrt{Q/n}} \sim \mathsf{T}(n),$ Student's t distribution with $n$ degrees of freedom.

Finally, if $Q_1 \sim \mathsf{Chisq}(n)$ and independently $Q_2 \sim \mathsf{Chisq}(m),$ then $F = \frac{Q_1/n}{Q_2/m} \sim \mathsf{F}(n,m),$ Snedecor's F distribution with numerator degrees of freedom $n$ and denominator degrees of freedom $m.$

$Z$ is a frequently used notation for $\mathsf{Norm}(0,1),$ the standard normal distribution. The letter $Q$ is my personal favorite notation for a chi-squared distribution.

For more details and graphs for various parameters, you can find Wikipedia pages on each of these distributions. You are on your own with $G$ and Morphet's $Q$.

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Here are histograms of 100,000 realizations of each of $\mathsf{Chisq}(4),\, \mathsf{T}(5), \, \mathsf{F}(15,20)$ generated as discussed above. Red curves are the density functions of these distributions. [Formal proofs use moment generating functions and multivariate transformations.]

Note: For archival purposes, the R code used to make the plots is shown below.

set.seed(330);  m = 10^5 
par(mfrow=c(1,3)) # enable three panels per plot
 z.0 = rnorm(m)    # generate standard normals (three lines)
 z.1 = rnorm(m); z.2 = rnorm(m); z.3 = rnorm(m); z.4 = rnorm(m)
 z.5 = rnorm(m); z.6 = rnorm(m); z.7 = rnorm(m); z.8 = rnorm(m); z.9 = rnorm(m)
 q.4 = z.1^2 + z.2^2 + z.3^2 + z.4^2  # CHISQ(4) from four indep NORM(0,1)
 q.5 = z.5^2 + z.6^2 + z.7^2 + z.8^2 + z.9^2 # CHISQ(5) from five
 hist(q.4, prob=T, br=30, col="skyblue2", main="CHISQ(4)")
   curve(dchisq(x,4), add=T, lwd=2, col="red")
 t = z.0/sqrt(q.5/5)  # T(5) from NORM(0,1) and CHISQ(5)
 hist(t, prob=T, br=50, col="skyblue2", main="T(5)", ylim=c(0,.4))
   curve(dt(x,5), add=T, lwd=2, col="red", n=10001)
 q.15 = rchisq(m, 15); q.20 = rchisq(m, 20) # CHISQ(15) & CHISQ(20) sim directly
 f = (q.15/15)/(q.20/20) # F(15, 20) from CHISQ(15) & CHISQ(20)
 hist(f, prob=T, br=30, col="skyblue2", main="F(15,20)", ylim=c(0,1))
   curve(df(x, 15, 20), add=T, lwd=2, col="red")
par(mfrow=c(1,1))