ISYE 6644 EXAMS SCRIPT LATEST QUESTIONS AND
ANSWERS SURE A+
✔✔If X 1 = 2, X 2 = − 2, and X 3 = 0 are i.i.d. realizations from a Nor(μ , σ^2)
distribution, what is the value of the maximum likelihood estimate for the variance σ^2? -
✔✔σ^2 = (n-1)/n *S^2
8/3
✔✔Suppose we observe the Pois( λ ) realizationsX 1 = 5 , X 2 = 9 and X 3 = 1. What is
the maximum likelihood estimate of λ? - ✔✔The likelihood function is
L(λ)
Thus,
ln ( L ( λ ) )
This implies that
(d /d λ )*ln ( L ( λ ) ) = − n + (∑ i = 1 n x i )/ λ .
Setting the derivative to 0 and solving yields λ ^ = x ¯ = 5. Duh! What a surprise!
✔✔Suppose that we have a number of observations from a Pois( λ ) distribution, and it
turns out that the MLE for λ is λ ^ = 5. What's the maximum likelihood estimate of Pr ( X
= 3 )? - ✔✔By invariance,
Pr ^ ( X = k ) = e − λ ^ λ ^ k k ! ,
so that we have Pr ^ ( X = 3 ) = e − x ¯ x ¯ k k ! = e − 5 5 3 3 ! = 0.1404.
✔✔Suppose that we observe X 1 = 5, X 2 = 9, and X 3 = 1. What's the method of
moments estimate of E [ X 2 ]? - ✔✔The MOM estimator for E[X^2] = ( 25 + 81 + 1 ) / 3
=
✔✔Consider the PRN's U 1 = 0.1 , U 2 = 0.9 , and U 3 = 0.2. Use Kolmogorov-Smirnov
with α = 0.05 to test to see if these numbers are indeed uniform. Do we ACCEPT or
REJECT uniformity? - ✔✔Accept
, Since D < D α , n, we ACCEPT uniformity (though it's kind of a joke since it's only based
on 3 observations)
✔✔What does GIGO mean? - ✔✔Garbage-in-garbage-out
✔✔What's a good distribution for modeling heights of people? - ✔✔Normal
✔✔What's a good distribution for modeling the number of random customer arrivals to a
store? - ✔✔The Poisson distribution is used for counting the number of arrivals over
some interval of time.
✔✔TRUE or FALSE? If the expected value of your estimator equals the parameter that
you're trying to estimate, then your estimator is unbiased. - ✔✔TRUE. (This is the
definition of unbiasedness in words.)
✔✔TRUE or FALSE? If X 1 , X 2 , . . . , X n are i.i.d. with mean μ, then the sample mean
X ¯ is unbiased for μ. - ✔✔TRUE
✔✔What is the MSE of an estimator? - ✔✔𝐵𝑖𝑎𝑠^2+𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒
✔✔Suppose that X 1 = 4 , X 2 = 3 , X 3 = 5 are i.i.d. realizations from an E x p ( λ )
distribution. What is the MLE of λ? - ✔✔Since λ^ = 1/ X ¯ = 0.25
✔✔TRUE or FALSE? It's possible to estimate two MLEs simultaneously, e.g., for the N
o r ( μ , σ 2 ) distribution. - ✔✔TRUE (it's possible based on taking the partial derivatives
of the likelihood function with respect to each parameter).
✔✔TRUE or FALSE? Sometimes it might be difficult to obtain an MLE in closed form. -
✔✔TRUE. (Think of the gamma example that we did.)
✔✔Suppose that the MLE for a parameter θ is θ ^ = 4. Find the MLE for θ. -
✔✔Invariance immediately implies that the MLE of sqrt(θ) is simply sqrt( θ ^) = 2θ ^ = 2
✔✔If you're in this course, then you should always love your. . . - ✔✔Method of
Moments !
✔✔TRUE or FALSE? If X 1 , X 2 , . . . , X n are i.i.d., then the MoM estimator for E [ X 3
]is 1 n ∑ i = 1 n X i 3. - ✔✔TRUE (by definition of MoM estimator)
✔✔Suppose H 0 is true, but you've just rejected it! What have you done? - ✔✔You've
committed a Type I error.
ANSWERS SURE A+
✔✔If X 1 = 2, X 2 = − 2, and X 3 = 0 are i.i.d. realizations from a Nor(μ , σ^2)
distribution, what is the value of the maximum likelihood estimate for the variance σ^2? -
✔✔σ^2 = (n-1)/n *S^2
8/3
✔✔Suppose we observe the Pois( λ ) realizationsX 1 = 5 , X 2 = 9 and X 3 = 1. What is
the maximum likelihood estimate of λ? - ✔✔The likelihood function is
L(λ)
Thus,
ln ( L ( λ ) )
This implies that
(d /d λ )*ln ( L ( λ ) ) = − n + (∑ i = 1 n x i )/ λ .
Setting the derivative to 0 and solving yields λ ^ = x ¯ = 5. Duh! What a surprise!
✔✔Suppose that we have a number of observations from a Pois( λ ) distribution, and it
turns out that the MLE for λ is λ ^ = 5. What's the maximum likelihood estimate of Pr ( X
= 3 )? - ✔✔By invariance,
Pr ^ ( X = k ) = e − λ ^ λ ^ k k ! ,
so that we have Pr ^ ( X = 3 ) = e − x ¯ x ¯ k k ! = e − 5 5 3 3 ! = 0.1404.
✔✔Suppose that we observe X 1 = 5, X 2 = 9, and X 3 = 1. What's the method of
moments estimate of E [ X 2 ]? - ✔✔The MOM estimator for E[X^2] = ( 25 + 81 + 1 ) / 3
=
✔✔Consider the PRN's U 1 = 0.1 , U 2 = 0.9 , and U 3 = 0.2. Use Kolmogorov-Smirnov
with α = 0.05 to test to see if these numbers are indeed uniform. Do we ACCEPT or
REJECT uniformity? - ✔✔Accept
, Since D < D α , n, we ACCEPT uniformity (though it's kind of a joke since it's only based
on 3 observations)
✔✔What does GIGO mean? - ✔✔Garbage-in-garbage-out
✔✔What's a good distribution for modeling heights of people? - ✔✔Normal
✔✔What's a good distribution for modeling the number of random customer arrivals to a
store? - ✔✔The Poisson distribution is used for counting the number of arrivals over
some interval of time.
✔✔TRUE or FALSE? If the expected value of your estimator equals the parameter that
you're trying to estimate, then your estimator is unbiased. - ✔✔TRUE. (This is the
definition of unbiasedness in words.)
✔✔TRUE or FALSE? If X 1 , X 2 , . . . , X n are i.i.d. with mean μ, then the sample mean
X ¯ is unbiased for μ. - ✔✔TRUE
✔✔What is the MSE of an estimator? - ✔✔𝐵𝑖𝑎𝑠^2+𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒
✔✔Suppose that X 1 = 4 , X 2 = 3 , X 3 = 5 are i.i.d. realizations from an E x p ( λ )
distribution. What is the MLE of λ? - ✔✔Since λ^ = 1/ X ¯ = 0.25
✔✔TRUE or FALSE? It's possible to estimate two MLEs simultaneously, e.g., for the N
o r ( μ , σ 2 ) distribution. - ✔✔TRUE (it's possible based on taking the partial derivatives
of the likelihood function with respect to each parameter).
✔✔TRUE or FALSE? Sometimes it might be difficult to obtain an MLE in closed form. -
✔✔TRUE. (Think of the gamma example that we did.)
✔✔Suppose that the MLE for a parameter θ is θ ^ = 4. Find the MLE for θ. -
✔✔Invariance immediately implies that the MLE of sqrt(θ) is simply sqrt( θ ^) = 2θ ^ = 2
✔✔If you're in this course, then you should always love your. . . - ✔✔Method of
Moments !
✔✔TRUE or FALSE? If X 1 , X 2 , . . . , X n are i.i.d., then the MoM estimator for E [ X 3
]is 1 n ∑ i = 1 n X i 3. - ✔✔TRUE (by definition of MoM estimator)
✔✔Suppose H 0 is true, but you've just rejected it! What have you done? - ✔✔You've
committed a Type I error.
