The Question details are in the images Consider the following joint pmf:рх,у (л, у) | у %3D 08/27y = 1Y = 2y = 34/278/27x = 04/272/27x = 1x = 21/27(a)Calculate the marginal pmfs.(b)Calculate E[X] and E[Y].(c)Calculate the pmf of XY.(d)Calculate E[XY].(e)Calculate Cov (X,Y).(f)Are X and Y independent random variables? Suppose we have a sample of 100 radioactive particles whose lifetimes are distributedExp(log(2)/B) for some unknown half-life B > 0.(a)X1, x2, . .. , x100 . Write the log-likelihood function for 3.Suppose we measure the time it takes for these particles to decay, obtaining the observations(b)What is BMLE in this setting?Estimate the distribution of BMLE using the CLT. Be sure to give any parameters for the ap-(c)proximating distribution.(d)If B = 3, what is the probability that|BMLE – 3| < 0.05?(e)another) and compute BMLE for each of these 7 samples, what is the average number of samples forwhich |BMLE – 3| < 0.05?If B = 3, and we draw 7 such samples of size 100 from this distribution (independently of oneConsider a population with unknown pdf, and parameters a > -1, 0 > 0. If X is arandom observation from this distribution, suppose that we have reason to believe the moments of X dependon the parameters in the following way:θ (α + 1)0² (a + 1)= E[X] =12 = E[X*] =a + 2a + 3(a)Show that the ratior =is just a quadratic polynomial in a. [A quadratic polynomial is just a function of the form p(x)ax? + bx + c for some constants a, b, c, like 5x² + 3x –2.](Ъ)intermediate calculations. This isn’t a requirement, just a suggestion.]Determine the method of moments estimators for a and 0. [Hint: Use the ratio r for your(c)second moment are given bySuppose we collect a sample from this population, and find that the sample mean and sample9=427m2 =5What are ô MoM, 0M0M, based on this sample data?Consider a random variable X with pdfI 0< x <1* 1< x < 2f (x) =otherwise.(a)Find the MGF of X.(b) (8 pts) Use the MGF to obtain the first four moments of X:E[X], E[X*], E[Xx³], E[X*).Hint: e' = Eok=0

Name: The Question details are in the images Consider the following joint pm
Uploaded: 2024-03-20T07:07:37.000Z
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Description: The Question details are in the images Consider the following joint pmf:рх,у (л, у) | у %3D 08/27y = 1Y = 2y = 34/278/27x = 04/272/27x = 1x = 21/27(a)Calculate the marginal pmfs.(b)Calculate E[X] and E[Y].(c)Calculate the pmf of XY.(d)Calculate E[XY]