I7
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
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
