Question

Let X1,X2,...,Xn be i.i.d. Geometric(θ), θ = 1,2,3,... random
variables.

a) Find the maximum likelihood estimator of θ.

b) In a certain hard video game, a player is confronted with a
series of AI opponents and has an θ probability of defeating each
one. Success with any opponent is independent of previous
encounters. Until ﬁrst win, the player continues to AI contest
opponents. Let X denote the number of opponents contested until the
player’s ﬁrst win. Suppose that data of 10 players was collected:
7,4,3,1,12,10,2,1,4,6

What is the MLE of the probability that a player contests ﬁve or
more AI opponents in a game until the ﬁrst win?

Answer #1

A player of a video game is confronted with a series of
opponents and has an 85% probability of defeating each one. Success
with an opponent is independent of previous encounters. Until
defeated, the player continues to contest opponents.
What is the probability mass function of a number of opponents
contested in a game?
What is the probability that a player defeats at least three
players in a game?
What is the expected number of opponents contested in a
game?...

. A player of a video game is confronted with a series of
opponents and has an 75% probability of defeating each one. Success
with any opponent is independent of previous encounters. Until
defeated, the player continues to contest opponents. (a) What is
the probability that a player defeats at least two opponents in a
game? (b) What is the expected number of opponents contested in a
game? (c) What is the probability that a player contests four or
more...

1. Find the maximum likelihood estimator (MLE) of θ based on a
random sample X1, . . . , Xn from each of the following
distributions
(a) f(x; θ) = θ(1 − θ) ^(x−1) , x = 1, 2, . . . ; 0 ≤ θ ≤ 1
(b) f(x; θ) = (θ + 1)x ^(−θ−2) , x > 1, θ > 0
(c) f(x; θ) = θ^2xe^(−θx) , x > 0, θ > 0

6. Let θ > 1 and let X1, X2, ..., Xn be a random sample from
the distribution with probability density function f(x; θ) =
1/(xlnθ) , 1 < x < θ.
a) Obtain the maximum likelihood estimator of θ, ˆθ.
b) Is ˆθ a consistent estimator of θ? Justify your answer.

let X1,X2,..............,Xn be a r.s from
N(θ,1). Find the best unbiased estimator for
(θ)^2

Let X1, X2, . . . , Xn be iid random variables with pdf
f(x|θ) = θx^(θ−1) , 0 < x < 1, θ > 0.
Is there an unbiased estimator of some function γ(θ), whose
variance attains the Cramer-Rao lower bound?

Let X1,X2, . . . ,Xn be a random sample of size n
from a geometric distribution for which p is the probability
of success.
(a) Find the maximum likelihood estimator of p (don't use method of
moment).
(b) Explain intuitively why your estimate makes good
sense.
(c) Use the following data to give a point estimate of p:
3 34 7 4 19 2 1 19 43 2
22 4 19 11 7 1 2 21 15 16

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