100 losses are independent, identically distributed, with a
common uniform distribution over (0, 6). Using the...
100 losses are independent, identically distributed, with a
common uniform distribution over (0, 6). Using the normal
approximation, calculate the probability that the sum of all the
losses (the aggregate loss) will exceed 275.
Let Z1, Z2, . . . , Zn be independent and identically
distributed as standard normal random variables. Prove the
distribution of ni=1 Zi2 ∼ χ2n.
Thanks!
Let ?1 , ?2 , ... , ?? be independent, identically distributed
random variables with p.d.f. ?(?) = ???−1, 0 ≤ ? ≤ 1 . c) Show that
the maximum likelihood estimator for ? is biased, and find a
function of the mle that is unbiased. (Hint: Show that the random
variable −ln (??) is exponential, the sum of exponentials is Gamma,
and the mean of 1/X for a gamma with parameters ? and ? is 1⁄(?(? −
1)).) d)...
Let X1, X2, . . . be a sequence of independent and identically
distributed random variables where the distribution is given by the
so-called zero-truncated Poisson distribution with probability mass
function; P(X = x) = λx/ (x!(eλ − 1)), x = 1, 2,
3...
Let N ∼ Binomial(n, 1−e^−λ ) be another random variable that is
independent of the Xi ’s.
1) Show that Y = X1
+X2 + ... + XN has a Poisson distribution
with mean nλ.
Let X1 and X2 be independent identically
distributed random variables with pmf p(0) = 1/4, p(1) = 1/2, p(2)
= 1/4
(a) What is the probability mass function (pmf) of X1
+ X2?
(b) What is the probability mass function (pmf) of
X(2) = max{X1, X2}?
(c) What is the MGF of X1?
(d) What is the MGF of X1 + X2
Let X1 and X2 be independent identically distributed random
variables with pmf p(0) = 1/4, p(1) = 1/2, p(2) = 1/4
(a) What is the probability mass function (pmf) of X1 + X2?
(b) What is the probability mass function (pmf) of X(2) =
max{X1, X2}?
(c) What is the MGF of X1?
(d) What is the MGF of X1 + X2? (Note: The formulas we did were
for the continuous case, so they don’t directly apply here, but you...
A charity receives 2025 contributions. Contributions are assumed
to be independent and identically distributed with mean 3125 and
standard deviation 250. Calculate the probability that the total
amount of contributions is greater than $6,320,000.
1. What do I know?
2. What do I want to find out?
3. What do we expect the answer to be?
4. How do I go from what I know to what I want to find?
5. Is the answer consistent with what I
expected?
You are given a number of i.i.d. (independent and identically
distributed) observations that are (continuously) uniformly
distributed in the interval from X to X+10 , where X is an unknown
real valued parameter. Derive the ML (maximum likelihood) estimator
for X. Given the observations 16.10 , 22.84 , 19.96 , 24.54 , 15.36
, 19.01 , 15.65 , 24.20 , 14.63 , 22.33 , compute the ML estimate
for X. If the ML estimate is a range of values, then...
You are given a number of i.i.d. (independent and identically
distributed) observations that are (continuously) uniformly
distributed in the interval from X to X+7 , where X is an unknown
real valued parameter. Derive the ML (maximum likelihood) estimator
for X. Given the observations 28.91 , 26.52 , 28.54 , 28.69 , 26.86
, 23.90 , 26.08 , 26.73 , 25.65 , 25.14 , 29.51 , 26.77 , compute
the ML estimate for X. If the ML estimate is a...
Losses covered by an insurance policy are uniform on [0, 2000].
An insurance company reimburses losses with a deductible of 700.
Calculate the difference between the first quartile and the third
quartile on the insurance company’s reimbursement.
Let ?1 and ?2 be two independent random variables with uniform
distribution on [0, 1].
1. Write down the joint cumulative distribution function and
joint probability density function of ?1 + ?2 and ?1?2