For the negative binomial pdf p X ( k ; p , r ) = ( k + r − 1 k ) ( 1 − p ) k p r , find the maximum likelihood estimator for p if r is known.
For the negative binomial pdf p X ( k ; p , r ) = ( k + r − 1 k ) ( 1 − p ) k p r , find the maximum likelihood estimator for p if r is known.
Solution Summary: The author calculates the maximum likelihood estimator for p. Let X is a random variable shows the given negative binomial distribution.
For the negative binomial pdf
p
X
(
k
;
p
,
r
)
=
(
k
+
r
−
1
k
)
(
1
−
p
)
k
p
r
, find the maximum likelihood estimator for
p
if
r
is known.
Definition Definition Probability of occurrence of a continuous random variable within a specified range. When the value of a random variable, Y, is evaluated at a point Y=y, then the probability distribution function gives the probability that Y will take a value less than or equal to y. The probability distribution function formula for random Variable Y following the normal distribution is: F(y) = P (Y ≤ y) The value of probability distribution function for random variable lies between 0 and 1.
If X₁, X₂,..., Xn are the values of a random sample from an exponential population, find the
maximum likelihood estimator of its parameter 0.
If the pdf of a random variable X is
f(x) = { 0.15-0.15(x-0.5), x ≥ 0.5
otherwise
Find the moment generating function and use it to find the mean and variance of X.
For the cumulative distribution function, find
a. U
b. P(x> 2)
c. F(x= 6)
d. P(x = 6)
e. f(x = 4)
f. f(x= 6)
g. F(x = 5.5)
h. F(4)
f. Variance of U
F(x) =<
0,
0.7,
0.9,
U₁
x < 1
1
Chapter 5 Solutions
An Introduction to Mathematical Statistics and Its Applications (6th Edition)
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