A manufacturing company produces many different products for wholesale distribution. Its flagship product, a reverse cylindrical dipolar tube, has a failure distribution given by the following PDF: 0.000192t² 0sts 25 years elsewhere f(t) = (a) Find R(15) and MTTF. (b) Find the probability that a failure will occur after the second year and by the end of the tenth year. (c) Find the median time to failure. (d) Construct the hazard rate function, At. Is it IFR, DRF, or CFR?

What was the maximum sustained yield under the logistic harvest model when k=500 and r=0.4? Values 1 500 1000 49L K L N N.d num [1:601] 0 123456789... num [1:50] 5 157.502 1079.678 0.732 24.054 ... num [1:50] 114 329 349 350 350 ... 0.4 int [1:50] 1 2 3 45678910 50 t Nt r Z Z L P T time y Files Plots Packages Help Viewer Presentation Zoom Export num [1:601] 0.4 0.399 0.398 0.398 0.397 ... Publish Observed Growth Rate 69 -0.1 0.1 0.3 0.5 Observed Growth Maximum Growth Carrying Capacity 0 100 200 300 400 500 600

DHL usually gives its customer a 60-minute window in which it expects home deliveries to be delivered. An independent consultant estimates that the final delivery time within the window is a delivery that follows the following condensation function:a. Draw the density function of the event.  b. What is the expected value of the transaction? c. What is the variance of the event?

Suppose that the interval between eruptions of a particular geyser can be modelled by an exponential distribution with an unknown parameter 0 > 0. The probability density function of this distribution is given by f(x; 0) = 0e 0¹, x > 0. The four most recent intervals between eruptions (in minutes) are x₁ = 32, x₂ = 10, x3 = 28, x4 = 60; their values are to be treated as a random sample from the exponential distribution. (a) Show that the likelihood of based on these data is given by L(0) 04-1306 = (b) Show that L'(0) is of the form L'(0) = 0³ e 1300 (4- 1300). (c) Show that the maximum likelihood estimate of 0 based on the data is ~ 0.0308 making your argument clear. (d) Explain in detail how the maximum likelihood estimate of that you have just obtained in part (c) relates to the maximum likelihood estimator of for an exponential distribution.

Consider independent observations y₁, ..., yn from the model Y₁ likelihood 1(µ) as appropriate, compute the following items. 1. Derive the maximum likelihood estimate û. 2. Write the second derivative of log-likelihood 1(μ). Poisson(μ). Using likelihood L(µ) and log-

Find the instantaneous form for each of the following phasors: 20 (a) A, = j10a, +a (b) B, = j4e-*a, + 6e*P*a, (c) C, = j2e=20%e¬J#lda -20z

2.11 Let X have the standard normal pdf, fx(x) = (1/√√2π) e¯2²/² (a) Find EX² directly, and then by using the pdf of Y = X2 from Example 2.1.7 and calculating EY. (b) Find the pdf of Y = |X|, and find its mean and variance.

Consider an MA(7) model with01= 1,02=-0.5,03= 0.25,04=-0.125,05= 0.0625,06=-0.03125, and07= 0.015625. Find a much simpler model that has nearly the samey-weights.