Reference no: EM132274477

Problem 1 - A random variable Y has probability density function given by:

a) Find k such that f(y) is a valid PDF.

b) Suppose Y measure the rate of leakage from a pipe. The leakage can be very costly and the monthly cost C, in dollars, can be estimated by C(Y) = 20Y² - 5. Find the average and variance in monthly cost.

c) Using your results from (b), give an approximate interval of costs that would occur at least 75% of the time.

d) Use an appropriate transformation method to derive the PDF for C(Y). Use this density to find the interval around the mean that would occur 75% of the time. This means that the remaining 25% probability is equally distributed beyond the lower and upper bounds of this interval.

Problem 2 - Transformations!

a) Suppose the time, W, it takes to complete a technical task at a workshop has probability density function

Using the appropriate transformation methods, find the density function for the average time it takes two workers to complete this technical task: S = (W₁ +W₂)/2.

b) Derive the moment generating function of a standard normal random variable. Use point form to explain each step in your derivation.

c) Use your standard normal MGF from (b) to derive the MGF of any normal random variable with mean μ and variance σ². (Hint: What is the connection between any N(μ, σ²) and N(0, 1)?)

d) Use the MGF from (c) to show that the distribution of the sample mean, X^-, of n random observations drawn from the same N(μ, σ²) distribution is:

X^- ∼ N (μ, σ²/n)

Where the sample mean is X^- = (_i=1∑ⁿX_i)/n.

Note that since X₁, X₂, . . . , X_n are random, X^-, which is a function of these n observations, will also be random.

Problem 3 - Among diabetics, the fasting blood glucose level G may be assumed to be approximately normally distributed with mean 106 milligrams per 100 millilitres (mg/100 ml) and standard deviation of 8 milligrams per 100 millilitres. Use and show probability notation for the problems below:

a) Without computing, under this distribution, do you think it will be very likely that a diabetic will have a fasting blood glucose level over 122 mg/100 ml? Briefly explain.

b) What is the 80^th percentile of fasting blood glucose levels of diabetics? Interpret this value.

c) What is the range of (the most common) fasting blood glucose levels of 80% of diabetics?

d) A diabetic is selected at random. Find the probability that their fasting blood glucose level is between 100 and 120 ml/100 ml.

e) Using your result from Problem 2, part (d), state the probability distribution of the average fasting blood glucose level of 3 randomly selected diabetics. What is the probability that their average fasting blood glucose level will exceed 110 mg/100 ml?

f) Do you think the probability that the average fasting blood glucose level exceeding 110 mg/100 ml from part (e) will be larger, smaller, or no different if instead we were examining the average blood glucose levels of 20 random selected diabetics? Explain your answer.

Problem 4 - The proportion of contaminant X and Y found among the contaminants of a soil sample have the following probability density function:

a) Sketch out the domain of the density function. Label all axes and boundaries.

b) Using your sketch in (a), carefully set up the limits of your integration to find the value of c that would make this a valid probability density function.

c) Determine whether the presence of contaminant X is independent of the presence of contaminant Y. Explain your conclusion.

d) If half of the contaminants in a tested soil sample were primarily contaminant X, what are the chances that Y makes up at least 25% of the contaminant in the soil sample? Show ALL your work, including any functions and relevant domains. You may find your graph in (a) helpful in determining any boundaries.

e) On average, what proportion of contaminant in a soil sample is made up of Y, if 50% is contaminant X?