Statistics / Probability and Statistical Inference 9 / Chapter 5.5 / Problem 12E

Probability and Statistical Inference | 9th Edition | ISBN: 9780321923271 | Authors: Robert V. Hogg, Elliot Tanis, Dale Zimmerman

Let the independent random variables X1 and X2 be N(0, 1)

Chapter 5, Problem 12E

(choose chapter or problem)

Let the independent random variables $X_{1}$ and $X_{2}$ be $N(0,1)$ and $\chi^{2}(r)$, respectively. Let $Y_{1}=X_{1} / \sqrt{X_{2} / r}$ and $Y_{2}=X_{2}$.

(a) Find the joint pdf of $Y_{1}$ and $Y_{2}$.

(b) Determine the marginal pdf of $Y_{1}$ and show that $Y_{1}$ has a $t$ distribution. (This is another, equivalent, way of finding the pdf of $T$.)

Equation Transcription:

${X}_{1}$

${X}_{2}$

$N(0,1)$

${\chi{}}^{2}(r)$

${Y}_{1}={X}_{1}/\sqrt{{X}_{2}/r}$

${Y}_{2}={X}_{2}$

${Y}_{1}$

${Y}_{2}$

$t$

$T$

Text Transcription:

X_1

X_2

N(0,1)

chi^2(r)

Y_1=X_1/ sqrt X_2/r

Y_2=X_2

Y_1

Y_2

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