Relationships among Normal, F, and t Distributions

Question

Let $X\sim N(14,9)$ and let $Y$ and $Z$ be standard Normal variables. Moreover, suppose that the three variables are independent.

a) Find the values of $c_1,c_2,c_3,c_4,c_5$ so that

$$[c_2(X+c_1)]^2/c_3(Y^2+Z^2)\sim F(c_4,c_5).$$

b) Consider now a sample of size $n=22$ from $X$ and denote by $\bar X$ and $S^2$ the corresponding sample mean and sample variance respectively. Provide information about the following probability $P(\bar X−14)/(S/√22).$

Guys, I don’t know how to solve it, any idea?

Answer 1

HINTS

• Remember that if you can define $X_1 \sim \chi^2(a)$ and $X_2 \sim \chi^2(b)$ then $$ \frac{X_1/a}{X_2/b} \sim F(a,b).$$ You may also want to recall that $\chi_i^2(k)$ is the sum of squares of $k$ standard normal distributions.

Answer 2

@gt6989b has given you good hints for (a) (+1).

(b) Here are facts about relevant distributions, most of which you should be able to find in your text. I will let you fill in some gaps and recall definitions of distributions. [Lower-case Greek letter $\nu$ ("nu") is often used to denote 'degrees of freedom'.]

Suppose $X_1, X_2, \dots, X_{22}$ is a random sample from $\mathsf{Norm}(\mu, \sigma).$

Then $\bar X \sim \mathsf{Norm}(\mu, \sigma/\sqrt{n})$ and $Z = \frac{\bar X - \mu}{\sigma/\sqrt{n}}\sim \mathsf{Norm}(0,1).$

Also, $\frac{(n-1)S^2}{\sigma^2} \sim \mathsf{Chisq}(\nu=n-1),$ so that $C^2 = \frac{S^2}{\sigma^2} \sim \frac{\mathsf{Chisq}(\nu=n-1)}{n-1}.$

Finally, $T = \frac{Z}{C} = \frac{\bar X - \mu}{S/\sqrt{n}} \sim \mathsf{T}(\nu = n-1).$