Let X be a negative binomial random variable with parameters n and p, and let Y be a binomial random variable with parameters n and p. Show that P { X > n } = P { Y < r } Hint: Either one could attempt an analytical proof of the preceding equation, which is equivalent to proving the identity ∑ i = n + 1 ∞ ( i − 1 r − 1 ) p r ( 1 − p ) i − r = ∑ i = 0 r − 1 ( n i ) × p i ( 1 − p ) n − i or one could attempt a proof that uses the probabilistic interpretation of these random variables. That is, in the latter case, start by considering a sequence of independent trials having a common probability p of success. Then try to express the events { X > n } and { Y < r } in terms of the outcomes of this sequence.
Let X be a negative binomial random variable with parameters n and p, and let Y be a binomial random variable with parameters n and p. Show that P { X > n } = P { Y < r } Hint: Either one could attempt an analytical proof of the preceding equation, which is equivalent to proving the identity ∑ i = n + 1 ∞ ( i − 1 r − 1 ) p r ( 1 − p ) i − r = ∑ i = 0 r − 1 ( n i ) × p i ( 1 − p ) n − i or one could attempt a proof that uses the probabilistic interpretation of these random variables. That is, in the latter case, start by considering a sequence of independent trials having a common probability p of success. Then try to express the events { X > n } and { Y < r } in terms of the outcomes of this sequence.
Solution Summary: The author explains that for a negative binomial random variable with parameters PX>n=PY, more than n variables should be required.
Let X be a negative binomial random variable with parameters n and p, and let Y be a binomial random variable with parameters n and p. Show that
P
{
X
>
n
}
=
P
{
Y
<
r
}
Hint: Either one could attempt an analytical proof of the preceding equation, which is equivalent to proving the identity
∑
i
=
n
+
1
∞
(
i
−
1
r
−
1
)
p
r
(
1
−
p
)
i
−
r
=
∑
i
=
0
r
−
1
(
n
i
)
×
p
i
(
1
−
p
)
n
−
i
or one could attempt a proof that uses the probabilistic interpretation of these random variables. That is, in the latter case, start by considering a sequence of independent trials having a common probability p of success. Then try to express the events
{
X
>
n
}
and
{
Y
<
r
}
in terms of the outcomes of this sequence.
The Martinezes are planning to refinance their home. The outstanding balance on their original loan is $150,000. Their finance company has offered them two options. (Assume there are no additional finance charges. Round your answers to the nearest cent.)
Option A: A fixed-rate mortgage at an interest rate of 4.5%/year compounded monthly, payable over a 30-year period in 360 equal monthly installments.Option B: A fixed-rate mortgage at an interest rate of 4.25%/year compounded monthly, payable over a 12-year period in 144 equal monthly installments.
(a) Find the monthly payment required to amortize each of these loans over the life of the loan.
option A $
option B $
(b) How much interest would the Martinezes save if they chose the 12-year mortgage instead of the 30-year mortgage?
The Martinezes are planning to refinance their home. The outstanding balance on their original loan is $150,000. Their finance company has offered them two options. (Assume there are no additional finance charges. Round your answers to the nearest cent.)
Option A: A fixed-rate mortgage at an interest rate of 4.5%/year compounded monthly, payable over a 30-year period in 360 equal monthly installments.Option B: A fixed-rate mortgage at an interest rate of 4.25%/year compounded monthly, payable over a 12-year period in 144 equal monthly installments.
(a) Find the monthly payment required to amortize each of these loans over the life of the loan.
option A $
option B $
(b) How much interest would the Martinezes save if they chose the 12-year mortgage instead of the 30-year mortgage?
When a tennis player serves, he gets two chances to serve in bounds. If he fails to do so twice, he loses the point. If he
attempts to serve an ace, he serves in bounds with probability 3/8.If he serves a lob, he serves in bounds with probability
7/8. If he serves an ace in bounds, he wins the point with probability 2/3. With an in-bounds lob, he wins the point with
probability 1/3. If the cost is '+1' for each point lost and '-1' for each point won, the problem is to determine the optimal
serving strategy to minimize the (long-run)expected average cost per point. (Hint: Let state 0 denote point over,two
serves to go on next point; and let state 1 denote one serve left.
(1). Formulate this problem as a Markov decision process by identifying the states and decisions and then finding the
Cik.
(2). Draw the corresponding state action diagram.
(3). List all possible (stationary deterministic) policies.
(4). For each policy, find the transition matrix and write an expression for the…
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