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Aug
25 |
comment |
What is known about the Gaussian measure of the unit ball in a Hilbert Space?
Or, the site : books.google.fr/… |
Aug
25 |
comment |
What is known about the Gaussian measure of the unit ball in a Hilbert Space?
For more detail, you can see the book of Michel Ledoux - "The Concentration of Measure Phenomenon. |
Aug
25 |
answered | What is known about the Gaussian measure of the unit ball in a Hilbert Space? |
Aug
24 |
answered | The density of x_1^n+x_2^n where x_i are Gaussian |
Aug
22 |
awarded | Teacher |
Aug
22 |
answered | Kernel width in Kernel density estimation |
Aug
21 |
answered | connection between the Gaussian and the Cauchy distribution |
Aug
21 |
comment |
connection between the Gaussian and the Cauchy distribution
Jon Peterson, you are correct. But why does it directly the calcul of cdf and pdf of $Y/X$, ie \begin{eqnarray*} F_Z(z)&=&\mathbb P(Z\leq z)=\mathbb P(Y/X\leq z)=\mathbb P(Y\leq zX)\\ &=&\mathbb P(Y\leq zX,\,X> 0)+ \mathbb P(Y\geq zX,\,X< 0),\,\, \mbox{that implies}\\ f_Z(z)&=& \frac{dF_Z(z)}{dz}=\int_{-\infty}^{+\infty}|x|f_Y(zx)f_X(x)\, dx\\ &=&\frac{1}{2\pi}\int_{-\infty}^{+\infty}|x|e^{-(z^2+1)x^2/2}\, dx=\frac{1}{\pi(x^2+1)}. \end{eqnarray*} The difficulty I encountered is how to prove that the characteristic function of the variable $ Y / X $ is the same as the Cauchy distribution ? |
Aug
21 |
answered | Estimating the mean of a truncated gaussian curve |