ENM 307 SAMPLE QUESTIONS
1. Probability Reminders Consider a (continuous) uniform random
variable in (0,2), fX (x) = 1/2 if 0 < x < 2 and fX (x) = 0 if otherwise.
a. Find E[X].
b. Find the cumulative distribution function of X, FX (x).
c. Find E[X 2 ].
2. Consider an exponential random variable with parameter 1, fX (x) =
e−x if x > 0 and fX (x) = 0 if otherwise.
a. Plot fX (x).
b. Find the cumulative distribution function of X, FX (x).
c. Find the median µ̃ of X. Note that the median has the following
property: FX (µ̃) = 1/2.
d. Find E[X].
3. You have data on daily sales of a slow-moving item. You think that
a Bernoulli distribution may be an appropriate fit. Recall that the
Bernoulli distribution has the following form: P (X = 0) = (1 − θ) and
P (X = 1) = θ.
Day
Sales
1
0
2
1
3
0
4
0
5
0
6
1
7
0
8
1
9
0
10
0
The goal is to estimate the unknown parameter θ using maximum likelihood estimation.
a. Find the likelihood function L(θ) of the above sample.
b. Find the log-likelihood function l(θ) of the sample
1
c. Differentiate l(θ) to find the maximum likelihood estimator (make
sure to check the second derivative).
4. Consider the following Random Number Generators. For each generator, generate three uniform random numbers, find the period and
observe whether it achieves the full cycle. If necessary, generate more
random variables to find the period of the generators.
a. (Zi = 3Zi−1 + 5) (mod(8)), Z0 = 1.
b. (Zi = 2Zi−1 + 3) (mod(8)), Z0 = 1.
c. (Zi = 11Zi−1 + 5) (mod(16)), Z0 = 1.
d. (Zi = 3Zi−1 + 2Zi−2 ) (mod(16)), Z0 = 1 Z1 = 5.
5. The following data is to be used as input in a simulation experiment.
You think the sample comes from a continuous uniform distribution in
(10,30) and would like to assess whether this would be the right fit.
Obs.
Value
1
26.24
2
10.44
3
13.37
4
28.96
5
19.05
6
20.51
7
15.37
8
24.05
9
15.49
10
25.23
a. Compare the empirical cumulative distribution with the theoretical
cumulative distribution filling in the below table.
x
Fn (x)
F̂ (x)
10.44
0.1
0.022
13.37
15.37
15.49
19.05
20.51
24.05
25.23
26.24
b. Do a Q-Q plot using the 0.25, 0.50 and 0.75 quantiles only.
c. Using five equal length intervals compute the χ2 test statistic. You
can use the following table.
Interval
Frequency (Observation)
Frequency (Theoretical)
(10,14)
2
2
(14,18)
(18,22)
(22,26)
(26,30)
28.96
1
d. Compute the Kolmogorov-Smirnov test statistic using the empirical
and theoretical cumulative distributions from part a.