Given below are 10 observations which we believe comes from the exponential distribution with λ= 10....

Given below are 10 observations which we believe comes from the exponential distribution with λ= 10.

Give the 10 ordered pairs which you would need to construct a probability plot you could use to verify this assumption.

1.08, 4.24, 33.40, 29.26, 2.94, 15.92, 1.96, 10.47, 16.71, 14.27

Expert Solution

density function in exponential distribution

lambda = 1/10

f(x) = lambda * exp( - lambda* x)

X	*Prob(X=x)*
1.08	0.089763
1.96	0.082201
2.94	0.074528
4.24	0.065442
10.47	0.035099
14.27	0.024003
15.92	0.020352
16.71	0.018806
29.26	0.005361
33.4	0.003544

Given below is the probaility density function

-------------------------------------------------------------------------------------

However, for taking lambda =10 ( which I suspect is wrong) the results hold and distribution plot seems to be not exponential

X	*Prob(X=x)*
1.08	0.000204
1.96	3.07E-08
2.94	1.71E-12
4.24	3.85E-18
10.47	3.38E-45
14.27	1.06E-61
15.92	7.25E-69
16.71	2.69E-72
29.26	8.4E-127
33.4	8.8E-145

orchestra answered 2 years ago

Use R and perform following: Generate 1000 observations from an exponential distribution with mean 10. Generate...

Use R and perform following: Generate 1000 observations from an exponential distribution with mean 10. Generate 1000 observations from a central t-distribution with 8 degree of freedom. Make a qqplot of observations in problem 1 versus quantiles generated from a t-distribution with 8 degree of freedom. Can the t distribution be used to approximate data in part 1?Submit the plot. Repeat above part but submit a qqplot of the observations in 1 versus quantiles from an exponential with mean 1....

The exponential distribution with rateλhas meanμ= 1/λ. Thus the method of moments estimator of λ is...

The exponential distribution with rateλhas meanμ= 1/λ. Thus the method of moments estimator of λ is 1/X. Use the following steps to verify that X is unbiased, but 1/X is biased. USING R: a) Generate 10000 samples of size n= 5 from the standard exponential distribution (i.e.λ= 1) using rexp(50000) and arranging the 50000 random numbers in a matrix with 5 rows. b) Use the apply() function to compute the 10000 sample means and store them in the object means....

The exponential distribution with rate λ has mean μ = 1/λ. Thus the method of moments...

The exponential distribution with rate λ has mean μ = 1/λ. Thus the method of moments estimator of λ is 1/X. Use the following steps to verify that X is unbiased, but 1/X is biased. a) Generate 10000 samples of size n = 5 from the standard exponential distribution (i.e. λ = 1) using rexp(50000) and arranging the 50000 random numbers in a matrix with 5 rows. b) Use the apply() function to compute the 10000 sample means and store...

For a random sample of sizenfrom an Exponential distribution with rate parameter λ (so that the...

For a random sample of sizenfrom an Exponential distribution with rate parameter λ (so that the density is fY(y) =λe−λy), derive the maximum likelihood estimator, the methods of moments estimator, and the Bayes estimator (that is, the posterior mean) using a prior proportional to λe−λ, for λ >0. (Hint: the posterior distribution will be a Gamma.)

Exponential distribution with λ = 0.5 Apply Chi-Square test to see whether the your hypothesized distribution...

Exponential distribution with λ = 0.5 Apply Chi-Square test to see whether the your hypothesized distribution fits your data. using excel

If Y is a random variable with exponential distribution with mean 1/λ, show that E (Y...

If Y is a random variable with exponential distribution with mean 1/λ, show that E (Y ^ k) = k! / (λ^k), k = 1,2, ...

Suppose the time to failure follows an exponential distribution with parameter λ. Based on type II...

Suppose the time to failure follows an exponential distribution with parameter λ. Based on type II censoring determine the likelihood and the maximum likelihood estimator for λ. Assume that n = 3 and the experiment ends after 2 failures.

a) Find the analytic MLE formula for exponential distribution exp(λ). Show that MLE is the same...

a) Find the analytic MLE formula for exponential distribution exp(λ). Show that MLE is the same as MoM estimator here. (b) A random sample of size 6 from the exp(λ) distribution results in observations: 1.636, 0.374, 0.534, 3.015, 0.932, 0.179. Find the MLE on this data set in two ways: by numerical optimization of the likelihood and by the analytic formula. For (b): please give both values from the analytic MLE formula and numerical MLE solution on this data set....

Suppose X1, ..., Xn are i.i.d. from an exponential distribution with mean θ. If we are...

Suppose X1, ..., Xn are i.i.d. from an exponential distribution with mean θ. If we are testing H0 : θ = θ0 vs Ha : θ > θ0. Suppose we reject H0 when ( X¯n/ θ0) > 1 + (1.645/ √n) (a) (10 points) Calculate the power function G(ζ). You may leave your answer in terms of the standard normal cdf Φ(x). (b) (5 points) Is this test consistent?

Consider two machines both of which have an exponential lifetime with rate λ. There is a...

Consider two machines both of which have an exponential lifetime with rate λ. There is a single repairman that can service machines at an exponential rate μ. – Set up the Kolmogorov backward equations in the matrix format P′(t) = RP(t). You do not need to solve the system. – Find the proportion of time that 0, 1, or 2 machines are down

Question