1. ## Probability Distributions

Consider a random sample Y1, . . . , Yn from a normal population with mean μ and with variance σ2. Define
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The question in my textbook then asks:For each of the random variables X1 , . . . , X5 , give its probability distribu-tion. (Do not forget to give the values of the parameters of the distribu-tion.)
I'm not sure what its asking, could I get some help starting off?

2. ## Re: Probability Distributions

1. $\displaystyle X_1=\frac{\overline{Y}-\mu}{\frac{\sigma}{\sqrt{n}}}$
Multiply the numerator and the denominator with a factor $\displaystyle n$ that gives us:
$\displaystyle X_1=\frac{n(\overline{Y}-\mu)}{n\left(\frac{\sigma}{\sqrt{n}}\right)}=\frac {[Y_1+\ldots+Y_n]-n\mu}{\sqrt{n}\sigma}$
It follows from the central limit theorem now that $\displaystyle X_1$ has a standard normal distribution.

2. $\displaystyle X_3=\frac{(n-1)S^2}{\sigma^2}$ has a chi square distribution with $\displaystyle n-1$ degrees of freedom.

3. If follows from $\displaystyle X_3$ that $\displaystyle S=\sqrt{\frac{X_3\sigma^2}{n-1}}$
If we substitute this in the epression for $\displaystyle X_2$ we obtain:
$\displaystyle \frac{\overline{Y}-\mu}{\frac{\sqrt{\frac{X_3\sigma^2}{n-1}}}{\sqrt{n}}}=\frac{X_1}{\sqrt{\frac{X_3}{n-1}}}$, with the information that $\displaystyle X_1 \sim N(0,1)$ and $\displaystyle X_3 \sim \chi^2_{n-1}$ if follows that $\displaystyle X_2$ has a student $\displaystyle t$ distribution with $\displaystyle n-1$ degrees of freedom.

4. $\displaystyle X_4=X_1^2$ and $\displaystyle X_1 \sim N(0,1)$ thus by definition $\displaystyle X_4 \sim \chi^2_{1}$

5. $\displaystyle X_5=X_1^2+X_3$, we know that $\displaystyle X_1 \sim N(0,1)$ and $\displaystyle X_3 \sim \chi^2_{n-1}$ thus $\displaystyle X_3=Z_1^2+\ldots+Z_{n-1}^2$ where $\displaystyle Z_i \sim N(0,1)$ (for each $\displaystyle i \in \{1,\ldots,n-1\}$) therefore $\displaystyle X_5 \sim \chi^2_{n}$