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Lecture notes of 3 pages for the course 3081 at Northeastern University (Homework answers)

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  • September 2, 2024
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MATH 3081 Homework 15 March 12, 2024
Name Max Langweil
5.2.1 A random sample of size 8-X1 = 1, X2 = 0, X3 = 1, X4 = 1, X5 = 0, X6 = 1, X7 = 1, and X8 = 0-is taken from
the probability function
k 1 k
PX (k; ✓) = ✓ (1 ✓) , k = 0, 1; 0 < ✓ < 1.
Find the maximum likelihood estimate for ✓.

L = F(X : ;G) :
3X :
4 -
G)N-Ex :



=
In 2 :
ExilnG + (n-Exilln(-e
"Vi

Alain
①E :


i =

-n

-
E :
I · 1 -
G


Exi GExi-En-GExi

⑳Es
S - = 0




④ Exi/n
:




5.2.3 Use the sample Y1 = 8.2, Y2 = 9.1, Y3 = 10.6, and Y4 = 4.9 to calculate the MLE for in the exponential pdf




fa (
y

-
x= fY (y; ) = e , y 0.
9 1
2)
10 6 4
:
= : +



I
.




L(X)
.
.


: Xe
Y


((x) [In In eiff
z
In = i =




a [Inx-xy) = a




[f si) o .
-




I =
y
:
St-Eyi
5.2.4 Suppose a random sample of size n is drawn from the probability model
✓2
✓2k e
PX (k; ✓) = , k = 0, 1, 2, . . .
k! n




Gigg
-no
ˆ
Find a formula for MLE, ✓.
i

C
Xi !
n




In L :
"Hnf-ne"
2
i = S K
-
i = I
Inki


(k)
d : zu



·ki-z ·




O
E



Eit

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