Reference no: EM131097525
HW 1: Due Monday 9:30am
1. MGF's: Here we derive MGF's and make some good use of them.
(a) Let random variable X, have an Exponential distribution with pdf : f(x) = βe-xβ, x > 0. Derive mgf of X, denoted as mX(t). Be sure to note the bounds on constant, t.
(b) If random variable, Y , has same distribution as in (b), what is distribution of S = X + Y (assuming the random variables are independent)?
2. Transformations: Univariate CDF and Direct Methods.
(a) Sec 3.9 pg 174: 1 (CDF and Direct Method)
(b) Let random variable W, have a Normal distribution with pdf :
f(x) = 1√2πσe- 12σ2(w-µ)2, -∞ < x < ∞.
Derive pdf of random variable, Y =W-µσ, via CDF and direct methods.
(c) Derive pdf of random variable, U = Y2, via CDF and direct methods.
(d) What are the names of the popular distributions found in (b) and (c)?
3. The waiting time Y until delivery of a textbook ordered from Amazon.com is uniformly distributed over the interval from 1 to 5 days. And of course, we wait until the quarter begins to order our textbook so we rent a book while we wait for our shipment to arrive. The cost of this delay is given by U = 2Y 2 + 3. Find the probability density function for U .
4. Let X and Y be random variables for which the joint pdf is as follows: fX,Y (x, y) = 2(x + y), 0 ≤ x ≤ y ≤ 1. Find the pdf of Z = X + Y .
5. Let X be the time that the server in a single-server queue will spend on a particular customer, and let Y be the rate at which the server can operate. A popular model for the conditional distribution of X given Y = y is to say fX|Y =y(x) = ye-xy, x > 0.
Let Y have some unknown pdf fY (y). The joint pdf of (X, Y )is then fX|Y =y(x)fY (y). Because 1/Y can be interpreted as the average service time, Z = XY measure how quickly, compared to the average, that the customer is served. For example, Z = 1 corresponds to an average service time, while Z > 1 means that the customer took longer than average, and Z < 1 means that the customer was served more quickly
than the average customer.
Given this problem it seems Z should depend on Y . Introduce dummy variable W = Y and derive the joint distribution (W, Z). Using the new joint pdf, determine if the random variables are independent or not.
|
Resulting implications for project management
: 1. Discuss the relationship among scope, schedule, and budget and the resulting implications for project management. 2. Discuss the criteria that should be used to make a project selection decision.
|
|
Which actions would be an appropriate fiscal policy response
: An increase in transfer payments combined with a decrease in government purchases would: If unemployment is the most significant problem in the economy, which of the following actions would be an appropriate fiscal policy response?
|
|
Description of the information system
: This is an information system assignment. The Instructors: Write and submit a detailed description of the information system chosen for the System Proposal.
|
|
Determining the mac process
: Briefly explain the MAC process and all of the frames that are used during this process.
|
|
An exponential distribution with pdf
: (a) Let random variable X, have an Exponential distribution with pdf : f(x) = βe-xβ, x > 0. Derive mgf of X, denoted as mX(t). Be sure to note the bounds on constant, t.
|
|
Aggregate demand or short-run aggregate supply
: List three variables that would cause a decrease in real GDP--indicate whether changes in each variable increase or decrease aggregate demand or short-run aggregate supply. Next, state three variables that would cause an increase in the price leve..
|
|
Questions about shortcoimgs of gdp
: What is the meaning of "GDP measures the size of the pie but not how the pie is divided?" Why is so important? Do you have clear examples to illustrate it?
|
|
Determine ? at which the system loses its ability to track
: Determine ω at which the system loses its ability to track the input. Is there anything that can be done to push this frequency higher? If so, test it by simulation
|
|
Procedure inevitably results in death
: One way to protect shellfish along the shoreline is to release large volumes of fresh water from the Mississippi River to flush oil out to sea. This procedure inevitably results in death to some of the saltwater shellfish while preventing more wid..
|