Hello,

I need assistance in explaining this output,

Let X be a exponential random variable with pdf f(x) = λe−λx for x > 0, and cumulative distribution function F(x).

(a) Show that F(x) = 1−e −λx for x > 0, and show that this function satisfies the requirements of a cdf (state what these are, and show that they are met). [4 marks]

(b) Draw f(x) and F(x) in separate graphs. Define, and identify F(x) in the graph of f(x), and vice versa. [Hint: write the mathematical relationships, and show graphically what the functions represent.] [4 marks]

(c) X has mgf M(t) = λ(λ−t) −1 . Derive the mean of the random variable from first principles (i.e. using the pdf and the definition of expectation). Also show how this mean can be obtained from the moment generating function. [10 marks]

(d)

(i) Show that F −1 (x) = − 1 λ ln(1 − x) for 0 < x < 1, where ln(x) is the natural logarithm. [4 marks]

(ii) If 0 < p < 1, solve F(xp) = p for xp, and explain what xp represents. [4 marks] (iii) If U ∼ U(0, 1) is a uniform random variable with cdf FU (x) = x (for 0 < x < 1), prove that X = − 1 λ ln(1 − U) is exponential with parameter λ. Hence, describe how observations of X can be simulated. [4 marks]

If I have two dice A and B, and I roll it twice, we have the outcomes A1, A2, B1, B2. Let X = A1+ A2, Y= B1+B2. What is the probability of X+Y <= 22.

Copier maintenance. The Tri-City Office Equipment Corporation sells an imported copier on a franchise basis and performs preventive maintenance and repair service on this copier. The data below have been collected from 45 recent calls on users to perform routine preventive maintenance service; for each call, X is the number of copiers serviced and Y is the total number of minutes spent by the service person. Assume that first-order regression model (1.1) is appropriate. (a) Obtain the estimated regression function. (b) Plot the estimated regression function and the data. How well does the estimated regression function fit the data? (c) Interpret b o in your estimated regression function. Does b o provide any relevant information here? Explain. (d) Obtain a point estimate of the mean service time when X = 5 copiers are serviced. Use R programming . The data set is 20 2 60 4 46 3 41 2 12 1 137 10 68 5 89 5 4 1 32 2 144 9 156 10 93 6 36 3 72 4 100 8 105 7 131 8 127 10 57 4 66 5 101 7 109 7 74 5 134 9 112 7 18 2 73 5 111 7 96 6 123 8 90 5 20 2 28 2 3 1 57 4 86 5 132 9 112 7 27 1 131 9 34 2 27 2 61 4 77 5

An important step in creating confidence intervals for proportions is to check whether the success/failure conditions have been met otherwise the interval created will not be valid (i.e. we should not have created that interval)!

The following examples are estimating the proportion of the population who likes Brussels sprouts. Try to determine whether or not the assumptions have been met.

1. A class has 15 girls and 10 boys. The teacher wants to form an unordered pair consisting
of 1 girl and 1 boy. How many ways are there to form such a pair?

2. For the same setup (i.e. class of 15 girls and 10 boys), the teacher wants to form an
unordered group of 3, consisting of 2 girls and 1 boy. How many ways are there to form
such a group?

3. For the same setup (i.e. class of 15 girls and 10 boys), assume the teacher now wants to
form an ordered group of 3, consisting of 2 girls and 1 boy (e.g., think of each student
having a different task, so their order, i.e. who does what, matters). How many ways
are there to form such a group?

Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 255 feet and a standard deviation of 37 feet. Let X be the distance in feet for a fly ball.

a. What is the distribution of X? X ~ N(___,__)

b. Find the probability that a randomly hit fly ball travels less than 278 feet.____ Round to 4 decimal places.

c. Find the 70th percentile for the distribution of distance of fly balls. Round to 2 decimal places. ___ feet

Fill in "___" please.

You are given a list of all employees. You group the names by department (Logistics, Sales, IT, Human Resource). Suppose you select all employees in Sales.

What type of sampling method did you use to select the employees? Explain your reasoning.

Im doing an econometric assignment and need to use the program STATA do estimate some linear regressions.

The dataset provided is the "natural log" of each variable.

What does the natural log mean? How is it calculated and how do I interoperate the data?

For example, a summary of the natural log of the unemployment rate shows:

Mean: -2.79

STD Deviation: 0.284

Min: -3.5833

Max: -1.9428

On average, indoor cats live to 15 years old with a standard deviation of 2.7 years. Suppose that the distribution is normal. Let X = the age at death of a randomly selected indoor cat. Round answers to 4 decimal places where possible.

a. What is the distribution of X? X ~ N(__,___)

b. Find the probability that an indoor cat dies when it is between 10.3 and 11.5 years old. ___

c. The middle 30% of indoor cats' age of death lies between what two numbers?
     Low: ____ years
     High: ____ years

Fill in the "___"

Suppose a fitness center has two weight-loss programs. Fifteen students complete Program A, and fifteen students complete Program B. Afterward, the mean and standard deviation of weight loss for each sample are computed (summarized below). What is the difference between the mean weight losses, among all students in the population? Answer with 95% confidence.

Prog A - Mean 10.5 St dev 5.6

Prog B - Mean 13.1 St dev 5.2

Applications that do not violate the OLS assumptions for inference. Identify the response and explanatory variable(s) for each problem. Write the OLS assumptions for inference in the context of each study.

1. Cricket Chirps. Researchers record the number of cricket chirps per minute and temperature during that time to investigate whether the number of chirps varies with the temperature.
2. Women’s Heights. A random selection of women aged 20-24 years are selected and their shoe size is used to predict their height

An article gave data on various characteristics of subdivisions that could be used in deciding whether to provide electrical power using overhead lines or underground lines. Here are the values of the variable x = total length of streets within a subdivision:

(a) Construct a stem-and-leaf display using the thousands digit as the stem and the hundreds digit as the leaf. (Enter numbers from smallest to largest separated by spaces. Enter NONE for stems with no values.)

What proportion of subdivisions have total length less than 2000? Between 2000 and 4000? (Round your answers to three decimal places.)

Components of a certain type are shipped to a supplier in batches of ten. Suppose that 50% of all such batches contain no defective components, 32% contain one defective component, and 18% contain two defective components. Two components from a batch are randomly selected and tested. What are the probabilities associated with 0, 1, and 2 defective components being in the batch under each of the following conditions? (Round your answers to four decimal places.)

(a) Neither tested component is defective.

(b) One of the two tested components is defective. [Hint: Draw a tree diagram with three first-generation branches for the three different types of batches.]