Question

X be random variable with distribution negative binomial NB(p,r) . r is known, 0<p<1 unknown. Find the UMVUE of log (p)

Answer #1

X be random variable with distribution negative binomial NB(p,r) . r is known, 0<p<1 unknown. The pmf is

We know that X is sufficient and complete for p. Let us assume the UMVUE of log(p) is g(X) where g fucntion statisfing E(g(X))= log(p). Then

Hence g(r) =0 and

This is the UMVUE of log(p).

X is a binomial random variable with n = 15 and p = 0.4.
a. Find using the binomial distribution.
b. Find using the normal approximation to the binomial
distribution.

If random variable X has a binomial distribution with n =8 and
P(success) = p =0.5, find the probability that X is at most 3.
(That is, find P(X ≤ 3))

Suppose that a random variable X has a binomial distribution
with n=2, p=0.5. Find the mean and variance of Y =
X2

The random variable X has a Binomial distribution with
parameters n = 9 and p = 0.7
Find these probabilities: (see Excel worksheet)
Round your answers to the nearest hundredth
P(X < 5)
P(X = 5)
P(X > 5)

True or False?
19. In a binomial distribution the random variable X is
discrete.
20. The standard deviation and mean are the same for the
standard normal distribution.
21. In a statistical study, the random variable X = 1, if the
house is colonial and X = 0 if the house is not colonial, then it
can be stated that the random variable is continuous. 22. For a
continuous distribution, P(X ≤ 10) is the same as P(X<10).
23. For...

Suppose X is binomial random variable with n = 18 and p = 0.5.
Since np ≥ 5 and n(1−p) ≥ 5, please use binomial distribution to
find the exact probabilities and their normal approximations. In
case you don’t remember the formula, for a binomial random variable
X ∼ Binomial(n, p), P(X = x) = n! x!(n−x)!p x (1 − p) n−x . (a) P(X
= 14). (b) P(X ≥ 1).

x is a binomial random variable with n=10 and p=.5. find the
probability of obtaining from 6 to 9 tails of a fair coin. use the
binomial probability distribution formula

Consider a random variable X with the following probability
distribution:
P(X=0) = 0.08, P(X=1) = 0.22,
P(X=2) = 0.25, P(X=3) = 0.25,
P(X=4) = 0.15, P(X=5) =
0.05
Find the expected value of X and the standard deviation of
X.

Let X be a binomial random variable with n =
8, p = 0.4. Find the following values. (Round your answers
to three decimal places.)
(a)
P(X = 4)
(b)
P(X ≤ 1)
(c)
P(X > 1)

A random variable XX with distribution
Exponential(λ)Exponential(λ) has the memory-less
property, i.e.,
P(X>r+t|X>r)=P(X>t) for all r≥0 and
t≥0.P(X>r+t|X>r)=P(X>t) for all r≥0 and t≥0.
A postal clerk spends with his or her customer has an
exponential distribution with a mean of 3 min3 min. Suppose a
customer has spent 2.5 min2.5 min with a postal clerk. What is the
probability that he or she will spend at least an additional 2 min2
min with the postal clerk?

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