## Question

###### 5. (16 points) Variations in the density in air can be measured by shining a laser...

5. (16 points) Variations in the density in air can be measured by shining a laser beam at a target on a surface. The square of the distance from the target satisfy a x distribution with density Jx(x) = -1/2-1/2-1/(20), 10. Here o >O is the standard deviation of the distance from the target of the image of the laser (a) Give the likelihood L(1, 29, 83, 84) based on four independent measurements 11,12,3,1, having the x? distribution above. (b) Find the maximum likelihood estimate ô for o. (c) Determine & for 1 = 0.091, x2 = 0.087, 63 = 0.094, and 14 = 0.106 square millimeters.

## Answers

Ansrer e ajo I) (a) Let, x,, X2, X3, X, one iis observations from the chinsonored distribution with put fx (x1) -112-112 -|(20) (osa So, the likelihood functionis, Lolx,,..,xn) f(xilo) 5 h 2 xi) ze 6(b) The log-lhelihood function is given by 1(012) -logt (ory) -2 logo Lino (50) n Is xi 12) Exi a) al(012) 121 is the solution of So, the MLE of dlo14) 2 2) 12 (C) If, x, = 0,091, 0.087, xz = 0.09h, Xy= 0.106, 2 – MIE(o)= 0.0945

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