1. Let X1,..., X, be an iid sample from Bernoulli(p). (a) Find the method of moment estimator of p. (b) Find the MLE for p.
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- Q. 25 If a Poisson variate X is such that P (x = 1) = P(x=2), Work out P (x = 4).Let X₁,..., Xn be a random sample from a distribution with one of two pdfs. If 0 = 1, then f(x;0=1) = 1 (0 < x < 1). If 0 = 2, then f(x;0= 2) = 2x1 (0 < x < 1). (a) Give a general form of the MLE of 0. (b) You are given a sample data of 3 values: 0.4, 0.5, 0.8. Find the MLE estimate of 0 based on the data.4. Question 4: Y is a discrete random variable with PMF S1/3, y=1 p(y) = 2/3, y = 2 (a) Find the MGF for Y. (b) Develop the first two raw moments from MGF and find the variance V(Y). Answers:
- 1. Let y1,.. , Yn be a random sample from a uniform distribution on the interval (0, B). Let B1 2 Li=1 Yi and B, = ) Max(y1,.., Yn) be estimators for B. Find the sampling distribution of B2 Show that B, and B, are unbiased estimators for B. 11.3. Let X1, .… , Xn be a random sample. Find the mle of 0 for the cases that the population pdf/pmf is, f(r;:0) = 0, %3D otherwiseSuppose you have a normally distributed population such that o? = 7. Using the sampling distribution of S?, we find that from a random sample of size n = 11, s² = 16.053. Does our population variance seem || reasonable? Note: We will consider our population variance reasonable if our x value falls within the interval x0.975 (df), x'o.025 (df)]. 1) Find the values of the interval [xo.975(df), x²o.025(df)]. x°0.975(df) = (Round to 3 decimals.) x'0.025(df) - (Round to 3 decimals.) 2) Find the value of x that comes from the data you collected in the problem statement. (Round to 3 decimals.) 3) Is the population variance reasonable? Yes O No
- - Let I = 1.135013 be the sample mean of an iid sample r1,..., x50 from a gamma population Gamma(1, 3). Here B > 0 is the unknown parameter of interest. Construct an approximate 95%-CI for B.3.3.24. Let X1, X2 be two independent random variables having gamma distribu- tions with parameters a₁ = 3, B₁ = 3 and a2 = 5, B2 = 1, respectively. (a) Find the mgf of Y = 2X₁ +6X2. (b) What is the distribution of Y?Let (x1, x2) be i.i.d. samples from a Poisson distribution with unknown parameter > 0. Suppose that we wish to test Ho : λ = 3 versus Ha: λ <3. (a) Consider the test which rejects for T = (x1 + x2₂) < 2. What is the level of significance a? (b) Find the power of this test for λ = 2.
- 6. Find the median of an exponential random variable X with parameter 1 = 4. (The median of X is m iff P(X < m) = 0.5.) Does it differ from E(X)? Try to explain.4.5. After fitting the regression model, y= Bo + Bix1 + Brx2+ B3x3+€ on 15 cases, it is found that the mean square error s? =3 and 0.5 0.3 0.2 0.6 0.3 6.0 0.5 0.4 (X'X)-' = 0.2 0.5 0.2 0.7 0.6 0.4 0.7 3.0 Find a. The estimate of V (ß1). b. The estimate of Cov(ß¡, ß3). c. The estimate of Corr(ß1, ß3). d. The estimate of V (ß¡ – B3).
