Question

The geometric distribution with parameter p represents the number of failures in a sequence of independent Bernoulli trials before a success occurs. Show that this distribution is of the exponential family.

Answer #1

Let {Xn} be a sequence of random variables that follow a
geometric distribution with parameter λ/n, where n > λ > 0.
Show that as n → ∞, Xn/n converges in distribution to an
exponential distribution with rate λ.

Match the distribution to the description
Group of answer choices
Bernoulli
Binomial
Geometric
Negative Binomial
Poisson
Counting the number of occurrences of an event in a continuous
interval
the sum of n independent bernoulli trials
given a series of independent bernoulli trials, stop when you
get r successes (where r can be any positive integer)
given a series of independent bernoulli trials, stop when you
get the first success ...

Consider a succession of independent Bernoulli tests of
parameter p. Let X be the number of failure before the first
success and Y the number of failure between the first success and
the second.
a) Calculate the joint density function of X and Y.
b) Calculate the conditional density function of X given Y =
y.
c) Calculate the conditional density function of Y given X =
x.
d) Are the variables X and Y independent? Argue your
answer.

Determine the CDF of Geometric Distribution with parameter
p.

5. If X and Y are independent geometric RVs with parameters p
and r re- spectively, show that U = min(X, Y ) is geometric with
parameter p + r − rp = 1−(1−r)(1−p).

Suppose Y_1, Y_2,… Y_n denote a random sample of a geometric distribution with parameter p. Find the maximum likelihood estimator for p.

Consider a sequence of independent trials of an experiment where
each trial can result in one of two possible outcomes, “Success” or
“Failure”. Suppose that the probability of success on any one trial
is p. Let X be the number of trials until the rth success is
observed, where r ≥ 1 is an integer.
(a) Derive the probability mass function (pmf) for X. Show your
work.
(b) Name the distribution by matching your resulting pmf up with
one in...

. Consider the Bernoulli distribution, P(X = x|p) = (p^x) (1 −
p) ^(1−x) for x = 0 and x = 1.
(a) Show that this is an exponential family.
(b) Find a sufficient statistic for p.
(c) Show that X is a m.v.u.e. for p.

Suppose that X|λ is an exponential random variable with
parameter λ and that λ|p is geometric with parameter p. Further
suppose that p is uniform between zero and one. Determine the pdf
for the random variable X and compute E(X).

a is a random variable which follows geometric distribution,
with parameter p = 0.1. b = ln(a)
1. what is the probability mass function of b.
2 Prob (b> ln(3)) = ?

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