Question

Two objects with unknown weights µ1 and µ2 are weighed separately on a scale and together (both placed on the scale), giving three measurements x1 = 15.6, x2 = 29.3 and x3 = 45.8. It is known from previous experience with the scale that measurements are independent and normally distributed about the true weights with variance = 1.

1. Determine the maximum likelihood estimates of µ1 and µ2.

Answer 1

Solution Given x 15.6 = 29.3 x3= 45.8 The weights are normally distributed. The pdf of a normal distribution is, (x-4 202 1 f(x)2o = The likelihood function ofui andu2 is, 1 - еxp 202 |(y)7 1 еxp 2 202 1 exp X .(т + 'n)- &x) 2a2 1 V2oexp (.f4--*)+ (%-x)+,(-+)E-} The log-likelihood function is: In(1(4.4.0)In ( еxp 3 2 Differentiate the log-likelihood:function with respect to/ui:

ln eIn(L(A.,o) )= (-2(-4)2(-4-) 0 (x-4(x-4-42) 24 61.4..(i) Differentiate the log-likelihood function with respect to uz: In (27o elIn(L(4,.0)) ((4--5) (4-3) (-)(2) (*#*}7]u]e 0(-2(4)-2(-4-4) 0 (x-4)x-4-42) 2u2 75.10....(ii) Solve (i) and (ii) simultaneously: Apply 2 x (ii) and (i) - (ii) 24 61.4 + = 24 150.2 (-) -24 88.8 44.4

Substitute 244.4 in (i) and solve for ui as follows 24 61.4 24444 61.4 24 61.4-44.4 2ц 3D17 14= 8.5 Therefore, the MLE ofu1 andu2 are 8.5 and 44.4 respectively