Xn is a Markov Chain with state-space
E = {0, 1, 2}, and transition matrix
0.4...
Xn is a Markov Chain with state-space
E = {0, 1, 2}, and transition matrix
0.4 0.2 ?
P = 0.6 0.3 ?
0.5 0.3 ?
And initial probability vector a = [0.2, 0.3,
?]
a) What are the missing values (?) in the transition matrix an
initial vector?
b) P(X1 = 0) =
c) P(X1 = 0|X0
= 2) =
d) P(X22 =
1|X20 = 2) =
e) E[X0] =
For the Markov Chain with state-space, initial vector, and...
6. Let X1, X2, ..., Xn be a random sample of a random variable X
from...
6. Let X1, X2, ..., Xn be a random sample of a random variable X
from a distribution with density
f (x) ( 1)x 0 ≤ x ≤ 1
where θ > -1. Obtain,
a) Method of Moments Estimator (MME) of parameter θ.
b) Maximum Likelihood Estimator (MLE) of parameter θ.
c) A random sample of size 5 yields data x1 = 0.92, x2 = 0.7, x3 =
0.65, x4 = 0.4 and x5 = 0.75. Compute ML Estimate...
Let n ≥ 2 and x1, x2, ..., xn > 0 be such that x1 +...
Let n ≥ 2 and x1, x2, ..., xn > 0 be such that x1 + x2 + · ·
· + xn = 1. Prove that √ x1 + √ x2 + · · · + √ xn /√ n − 1 ≤ x1/ √
1 − x1 + x2/ √ 1 − x2 + · · · + xn/ √ 1 − xn
) Let α be a fixed positive real number, α > 0. For a
sequence {xn},...
) Let α be a fixed positive real number, α > 0. For a
sequence {xn}, let x1 > √ α, and define x2, x3, x4, · · · by the
following recurrence relation xn+1 = 1 2 xn + α xn (a) Prove that
{xn} decreases monotonically (in other words, xn+1 − xn ≤ 0 for all
n). (b) Prove that {xn} is bounded from below. (Hint: use proof by
induction to show xn > √ α for all...
Let X = ( X1, X2, X3, ,,,, Xn ) is iid,
f(x, a, b) =...
Let X = ( X1, X2, X3, ,,,, Xn ) is iid,
f(x, a, b) = 1/ab * (x/a)^{(1-b)/b} 0 <= x <= a ,,,,, b
< 1
then,
Show the density of the statistic T = X(n) is given by
FX(n) (x) = n/ab * (x/a)^{n/(b-1}} for 0 <= x <=
a ; otherwise zero.
# using the following
P (X(n) < x ) = P (X1 < x, X2 < x, ,,,,,,,,, Xn < x
),
Then assume...
1. Consider the Markov chain {Xn|n ≥ 0} associated with
Gambler’s ruin with m = 3....
1. Consider the Markov chain {Xn|n ≥ 0} associated with
Gambler’s ruin with m = 3. Find the probability of ruin given X0 =
i ∈ {0, 1, 2, 3}
2 Let {Xn|n ≥ 0} be a simple random walk on an undirected graph
(V, E) where V = {1, 2, 3, 4, 5, 6, 7} and E = {{1, 2}, {1, 3}, {1,
6}, {2, 4}, {4, 6}, {3, 5}, {5, 7}}. Let X0 ∼ µ0 where µ0({i}) =...
The transition probability matrix of a Markov chain {Xn }, n =
1,2,3……. having 3
states...
The transition probability matrix of a Markov chain {Xn }, n =
1,2,3……. having 3
states 1, 2, 3 is P =
0.1 0.5 0.4
0.6 0.2 0.2
0.3 0.4 0.3
* and the initial distribution is P(0) = (0.7, 0.2,0.1)
Find:
i. P { X3 =2, X2 =3, X1 = 3, X0 = 2}
ii. P { X3 =3, X2 =1, X1 = 2, X0 = 1}
iii. P{X2 = 3}
Let Ω = {0, 1}^3 , that is, all possible (ordered) triples of
zeros and ones....
Let Ω = {0, 1}^3 , that is, all possible (ordered) triples of
zeros and ones.
Suppose that all outcomes have equal probability.
We define three random variables X1, X2, and X3 on this space
representing the first, second, and third digit, respectively.
We also define X = X1 + X2 + X3.
(i) Compute the values (across Ω) of each of the following
random variables: E(X|X1), E(E(X|X1)|X2), E(X2|X).
(ii) What is the probability mass function of E(X2|X).
Let X1, X2, ..., Xn be a random sample from a distribution with
probability density function...
Let X1, X2, ..., Xn be a random sample from a distribution with
probability density function f(x; θ) = (θ 4/6)x 3 e −θx if 0 < x
< ∞ and 0 otherwise where θ > 0
. a. Justify the claim that Y = X1 + X2 + ... + Xn is a complete
sufficient statistic for θ. b. Compute E(1/Y ) and find the
function of Y which is the unique minimum variance unbiased
estimator of θ.
b. Compute...