- Set the format type to Mac OS Extended (Journaled.) Click on the Apply button and wait until the process has completed. Select the volume you just created (this is the sub-entry under the drive entry) from the left side list. Click on the Erase tab in the DU main window. Set the format type to Mac OS Extended (Journaled.).
- Math was Mathematica's first great application area—and building on that success, Mathematica has systematically expanded into a vast range of areas, covering all forms of technical computing and beyond.
If you mean in everyday life,? Here are a couple. Say you need to measure a diagonal of some rectangular space, but for one reason or another some objects are in the way, or it's too long for a tape measure. If your Mac with Apple silicon is stuck on this screen, please contact Apple Support. Press and hold the power button for up to 10 seconds, until your Mac turns off. Try using safe mode to identify the issue. If the issue persists, unplug all accessories from your Mac, including printers, drives, USB hubs, and other nonessential devices.
We'll now turn our attention towards applying the theorem and corollary of the previous page to the case in which we have a function involving a sum of independent chi-square random variables. The following theorem is often referred to as the 'additive property of independent chi-squares.'
Theorem Section
Let (X_i) denote (n) independent random variables that follow these chi-square distributions:
- (X_1 sim chi^2(r_1))
- (X_2 sim chi^2(r_2))
- (vdots)
- (X_n sim chi^2(r_n))
Then, the sum of the random variables:
(Y=X_1+X_2+cdots+X_n)
follows a chi-square distribution with (r_1+r_2+ldots+r_n) degrees of freedom. That is:
(Ysim chi^2(r_1+r_2+cdots+r_n))
Proof
We have shown that (M_Y(t)) is the moment-generating function of a chi-square random variable with (r_1+r_2+ldots+r_n) degrees of freedom. That is:
(Ysim chi^2(r_1+r_2+cdots+r_n))as was to be shown.
Theorem Section
Let (Z_1, Z_2, ldots, Z_n) have standard normal distributions, (N(0,1)). If these random variables are independent, then:
(W=Z^2_1+Z^2_2+cdots+Z^2_n)
follows a (chi^2(n)) distribution.
Proof
Recall that if (Z_isim N(0,1)), then (Z_i^2sim chi^2(1)) for (i=1, 2, ldots, n). Then, by the additive property of independent chi-squares:
(W=Z^2_1+Z^2_2+cdots+Z^2_n sim chi^2(1+1+cdots+1)=chi^2(n))
That is, (Wsim chi^2(n)), as was to be proved.
Corollary Section
If (X_1, X_2, ldots, X_n) are independent normal random variables with different means and variances, that is:
(X_i sim N(mu_i,sigma^2_i))
for (i=1, 2, ldots, n). Then:
(W=sumlimits_{i=1}^n dfrac{(X_i-mu_i)^2}{sigma^2_i} sim chi^2(n))
Proof
Recall that:
(Z_i=dfrac{(X_i-mu_i)}{sigma_i} sim N(0,1))
Therefore:
(W=sumlimits_{i=1}^n Z^2_i=sumlimits_{i=1}^n dfrac{(X_i-mu_i)^2}{sigma^2_i} sim chi^2(n))
as was to be proved.
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