3. Consider the Weibull distribution with parameters ? and ? and the log logistic distribution with parameters ? and ? (refer to the class notes). Determine the probability density function (p.d.f.), survival function (s.f.) and hazard function of the distribution in each family that has median equal to 100 and 90th percentile equal to 300. Plot the p.d.f and the hazard function for each specific distribution. Compare and comment on their shapes. (Please use R or SAS to make the plots.)
Added by Raul V.
Close
Step 1
In R, generate the Weibull distribution with parameters and A: > Weibull(100,300) Show more…
Show all steps
Your feedback will help us improve your experience
Sri K and 86 other Intro Stats / AP Statistics educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
3. Consider the Weibull distribution with parameters ̢ and ̠ and the log logistic distribution with parameters ̤ and ̣ (refer to the class notes). Determine the probability density function (p.d.f.), survival function (s.f.) and hazard function of the distribution in each family that has median equal to 100 and 90th percentile equal to 300. Plot the p.d.f and the hazard function for each specific distribution. Compare and comment on their shapes. (Please use R or SAS to make the plots.)
Sri K.
2. Let X, Y and Z be independent random variables whose distributions are given by: X ~ Bin(25, 0.75), Y ~ Geo(0.5), Z ~ Po(5). a. Plot the probability mass functions outlined above in ONE figure with three subplots, using the same x-axis range for all of the subplots. b. Find a combination of X, Y and Z whose expectation is e^2 and whose variance is sqrt(pi). c. Find a combination of X, Y and Z whose expectation is equal to the variance. d. Compute the covariance of the resulting variable in part (c) with Z.
4.5. The exponential distribution has the probability function f(y; λ) = λ * exp(-αy), (4.37) for λ > 0 and y > 0. Consider estimating the mean λ for the exponential distribution based on a sample of n, Y1, Y2, ..., Yn. Determine the likelihood function and the log-likelihood function. Find the score function. Using the score function, find the maximum likelihood estimate (MLE) of λ. Find the observed and expected information for λ. Show that the standard error for λ is se(λ) = √(1/(nI(λ))). Show that the Wald test statistic for testing Ho: λ = 1 is W = (λ-1)^2/(1/(nI(λ))). Show that the score test statistic for testing Ho: λ = 1 is S = n(λ-1)^2/(I(λ)). Show that the likelihood ratio test statistic for testing Ho: λ = 1 is G = -2ln(L(1)/L(λ)). Plot W, S, and G for values of λ between 0.5 and 2 for n = 100. Comment on the results.
Recommended Textbooks
Elementary Statistics a Step by Step Approach
The Practice of Statistics for AP
Introductory Statistics
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD