The game of craps is played as follows: A player rolls two dice. If the sum of the dice is either a 2, 3, or 12, the player loses: if the sum is either a 7 or an 11, the player wins. If the outcome is anything else, the player continues to roll the dice until she rolls either the initial outcome or a 7. If the 7 comes first, the player loses, whereas if the initial outcome reoccurs before the 7 appears, the player wins. Compute the probability of a player winning at craps. Hint: Let E denote the event that the initial outcome is i and the player wins. The desired probability is ∑ i = 2 12 P ( E i ) . To compute P ( E i ) , define the events E i , n , to be the event that the initial sum is i and the player wins on the nth roll. Argue that P ( E i ) = ∑ n = 1 ∞ P ( E i , n ) .
The game of craps is played as follows: A player rolls two dice. If the sum of the dice is either a 2, 3, or 12, the player loses: if the sum is either a 7 or an 11, the player wins. If the outcome is anything else, the player continues to roll the dice until she rolls either the initial outcome or a 7. If the 7 comes first, the player loses, whereas if the initial outcome reoccurs before the 7 appears, the player wins. Compute the probability of a player winning at craps. Hint: Let E denote the event that the initial outcome is i and the player wins. The desired probability is ∑ i = 2 12 P ( E i ) . To compute P ( E i ) , define the events E i , n , to be the event that the initial sum is i and the player wins on the nth roll. Argue that P ( E i ) = ∑ n = 1 ∞ P ( E i , n ) .
The game of craps is played as follows: A player rolls two dice. If the sum of the dice is either a 2, 3, or 12, the player loses: if the sum is either a 7 or an 11, the player wins. If the outcome is anything else, the player continues to roll the dice until she rolls either the initial outcome or a 7. If the 7 comes first, the player loses, whereas if the initial outcome reoccurs before the 7 appears, the player wins. Compute the probability of a player winning at craps.
Hint: Let E denote the event that the initial outcome is i and the player wins. The desired probability is
∑
i
=
2
12
P
(
E
i
)
. To compute
P
(
E
i
)
, define the events
E
i
,
n
, to be the event that the initial sum is i and the player wins on the nth roll. Argue that
P
(
E
i
)
=
∑
n
=
1
∞
P
(
E
i
,
n
)
.
Question 1: Let X be a random variable with p.m.f
(|x| +1)²
x= -2, -1, 0, 1,2
f(x) =
C
0,
O.W
1. The value of c.
2. The c.d.f.
3. E(X).
4. E(2x+3).
5. E(X²).
6. E(3x²+4).
7. E(X(3X+4)).
8. Var(X).
9. Var (6-3X).
10. Find the m.g.f of the random variable X
Please could you explain how to do integration by parts for this question in detail please
2. Claim events on a portfolio of insurance policies follow a Poisson process with parameter
A. Individual claim amounts follow a distribution X with density:
f(x)=0.0122re001, g>0.
The insurance company calculates premiums using a premium loading of 45%.
(a) Derive the moment generating function Mx(t).
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