Extra Problems

Let's use Newton's method with x_0=1 to find an approximate solution to x^2−2=0. After 2 iterations, what is the approximate value of x_2? Use 3 decimal places.

1.417

Suppose X and Y have joint pdf f(x,y) = ½ for 0 ≤ x ≤1,0 ≤ y ≤ 2 find Pr(Y > X)

3/4

YES or NO? Consider again the joint p.d.f. from the previous question f(x,y) = 1/2 for 0 ≤ x ≤ 1, 0 ≤ y ≤ 2. Are X and Y independent random variables?

Yes

If 𝑋 and Y are i.i.d. Nor(1,4) random variables, find Var(3X−2Y).

52

If 𝑋 and 𝑌 are i.i.d. Unif(0,1) random variables, find E[2XY].

1/2

If 𝑍_1 , 𝑍_2 , … , 𝑍_2𝑛 are i.i.d. Nor(0,1), what is the distribution of ∑Z2_i ​ ?

Chi-Squared(χ2n2​)

If X_1, X_2, X_3, X_4 are iid with Exp(λ) with X_1 = 4, X_2 = 4, X_3 = 5, and X_4 = 10 what is the maximum likelihood of the estimator for λ

1/5.75

Suppose U1​ and U2​ are i.i.d. Unif(0,1) and let

X= √−2ln(U1​)​cos(2πU2​)

Find Pr(X>0.80).

0.21

Suppose you observe six i.i.d. Bern(p)realizations X_1=0, X_2=1, X_3​=0, X_4​=1, X_5​=0, and X_6=1. What's the maximum likelihood estimate of ln(p)?

-0.69

If U1,U2,U3​ are i.i.d. Unif(0,1), what is the distribution of −4ln⁡[(1−U1)U2U3]?

Erlang_3(0.25)

Consider a nonhomogeneous arrival process with rate function λ(t)=1+t/2 for t ≥ 0. Find the probability that there will be at most three arrivals before time t=2.

0.65

Suppose X1,…,X100​ are i.i.d. from an Exp(1) distribution. Use the Central Limit Theorem to find an approximate distribution of the summation ∑i=1100Xi​ and then calculate Pr(∑i=1100​Xi​>94) for me.

0.73

Suppose X1​,…,X100​ are i.i.d. from an Exp(1) distribution. What is the exact distribution of the summation ∑i=1100​Xi​?

Erlang_100(1)

TRUE or FALSE? An unbiased estimator does not necessarily have the lowest associated Mean Squared Error (MSE) among all estimators

TRUE

TRUE or FALSE? When conducting a X² goodness-of-fit test, a large value of the test statistic X²_0 (which can be calculated from observed data) indicates a bad fit

TRUE

TRUE or FALSE? Even though the exponential distribution is a special case of the gamma distribution, it is still possible to reject a goodness-of-fit test for the gamma, yet fail to reject it for the exponential, as the number of unknown parameters to be estimated may be different

TRUE

TRUE or FALSE? The Thinning Algorithm is typically used to generate a sequence of independent Bernoulli arrivals

FALSE

TRUE or FALSE? The ARENA expression to generate an exponential random variable with variance 4 is EXPO(4)

FALSE

Consider a particular data set of 10000 stationary waiting times obtained from a large queueing system. Suppose your goal is to get a confidence interval for the unknown mean. Would you rather use (a) 10 batches of 1000 observations or (b) 1000 batches of 10 observations each?

a.10 batches of 1000 observations

Suppose we're conducting a X² goodness-of-fit test to determine whether or not 100 i.i.d. observations are from a Bieber distribution. (The Bieber has a very, very ugly-looking p.d.f.) Suppose, after we divide the observations into various intervals and deal with the various unknown parameters, it turns out that the test has 6 degrees of freedom. Further suppose that we calculate a g-o-f statistic of X²_0 = 10.5. Will we ACCEPT (i.e., fail to reject) or REJECT the null hypothesis of the Bieber distribution? Use level of significance α = 0.10 for your test.

Accept

Consider the following 12 observations arising from a simulation:

154 180 175 162 200 173 191 183 193 112 169 166

Use the method of batch means to calculate a two-sided 90% confidence interval for the mean μ. In particular, use four batches of size three.

[151.58, 191.42]

Which of the following is best suited for a Single-Stage Normal means procedure?

Identifying which inventory policy produces the highest average profit

TRUE or FALSE? In the Single-Stage Normal means selection procedure, decreasing our indifference zone, δ∗ , while maintaining all other selection criteria increases the number of required samples, n

True

Suppose we are conducting a single-stage Bernoulli selection procedure to pick the best from among three treatments, X1​,X2​, and X3​. Based on our selection criteria, we set out to collect 184 samples from each option. Suppose that after collecting 154 samples, we have the following: X_1154 = 64, X_2154, X_3154 = 94. Which of the following is the best course of action?

cease all sample collection and choose the third population X_3

Suppose Z1 = 0.2, Z2 = 0.5 are i.i.d. Nor(0,1). Use the random vector Z = (0.2, 0.5), to generate a pair of observations X1, X2 coming from a bivariate normal distribution with mean vector μ = (2, 1) and covariance matrix Σ = [[3, 2], [2, 5]]. To make your computations easier, you can use the lower triangular Cholesky matrix C below (recall that CC^T = Σ).

C = [[sqrt(σ11), 0], [σ12/sqrt(σ11), sqrt(σ22 - σ12²/σ11)]] = [[sqrt(3), 0], [2/sqrt(3), sqrt(11/3)]] = [[1.73, 0], [1.15, 1.91]]

X_1 = 2.35, X_2 = 2.19

Let B(t) denote a Brownian bridge at time t. Let A represent the area under the process from time t = 0 to t = 1. From the lecture notes, we have:

A = ∫₀¹ B(t) dt, where A ~ Norm(0, 1/12)

Find the probability that A ≤ 0.5

0.96

TRUE or FALSE? In ARENA, it is only possible to model machine breakdowns (failures) by time, i.e., a failure can only occur after a certain amount of time.

False

TRUE or FALSE? In ARENA, two of the seize selection rules for resource sets are 'cyclical' and 'specific member'.

True