Problem 2 (10 points) Let X,, Xn be i.i.d. with one of two probability density functions....

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Problem 2 (10 points) Let X,, Xn be i.i.d. with one of two probability density functions. If 0 = 0, then 1 if 0 < x < 1 f(x | 0) 0 otherwise, while if = 1, then f(x | 0) = {1/(2/2) (1/(2) if 0 < x <1 otherwise, Find the MLE of 0. Continuous name range density f(x) for r range c.d.f. F(r) Mean for r range undefined. Variance 1 a+b (b-a) uniform (a,b) (a.b) b-a b-a 2 12 1 normal (0.1) (-x.x) (r) 0 1 normal (.02) 1 (-x.x) - /(x-4)/0 $> 2 " 02 exponential (A) gamma (1.A) (0.x) de-drz 1-e-Az 1/X 1/12 gamma (r.A) (0.x) T(r)-A-e-Ar 1-e-Ar (Ax)* k! r/A k=0 /A chi-square (n) gamma () for integer r (0.0) (1) (1) as above for A=. r = if n is even " 211 Rayleigh (0.0) re- 1-- (r+s)-(1-x)*-1 see Exercise 4.6.5 for integer r and s beta (r.s) (0.1) F(r)r(s) Cauchy (-x.x) arcsine -beta (1/2,1/2) (0.1) 1 1 1 + arctan(r) (1+x) 2 2k arcsin() 12 1- T 2 r rs r+8 (r+s)2(r+s+1) 2 12 11X Problem 2 (10 points) Let X,, Xn be i.i.d. with one of two probability density functions. If 0 = 0, then 1 if 0 < x < 1 f(x | 0) 0 otherwise, while if = 1, then f(x | 0) = {1/(2/2) (1/(2) if 0 < x <1 otherwise, Find the MLE of 0. Continuous name range density f(x) for r range c.d.f. F(r) Mean for r range undefined. Variance 1 a+b (b-a) uniform (a,b) (a.b) b-a b-a 2 12 1 normal (0.1) (-x.x) (r) 0 1 normal (.02) 1 (-x.x) - /(x-4)/0 $> 2 " 02 exponential (A) gamma (1.A) (0.x) de-drz 1-e-Az 1/X 1/12 gamma (r.A) (0.x) T(r)-A-e-Ar 1-e-Ar (Ax)* k! r/A k=0 /A chi-square (n) gamma () for integer r (0.0) (1) (1) as above for A=. r = if n is even " 211 Rayleigh (0.0) re- 1-- (r+s)-(1-x)*-1 see Exercise 4.6.5 for integer r and s beta (r.s) (0.1) F(r)r(s) Cauchy (-x.x) arcsine -beta (1/2,1/2) (0.1) 1 1 1 + arctan(r) (1+x) 2 2k arcsin() 12 1- T 2 r rs r+8 (r+s)2(r+s+1) 2 12 11X