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Mathematical Statistics (STAT3001/7301) Semester 1 2011 Assignment 1 (worth 10%) Please hand in your solutions to these problems to the lecturer by Friday 25/3/2011. 1. Suppose that X1; : : : ;X10 is a random sample from the distribution having pdf f(x) = x=2; 0 6 x 6 2; 0; elsewhere: What is the probability that all observations are greater than 1=2? Also nd the probability that at least one observation is less than 1/2. 2. Let X1 Binomial(n1; p) and X2 Binomial(n2; p) be independent for some n1; n2 2 Z+; p 2 [0; 1]. Let Y = X1+X2.

Mathematical Statistics (STAT3001/7301) Semester 1 2011 Assignment 1 (worth 10%) Please hand in your solutions to these problems to the lecturer by Friday 25/3/2011. 1. Suppose that X1; : : : ;X10 is a random sample from the distribution having pdf f(x) = x=2; 0 6 x 6 2; 0; elsewhere: What is the probability that all observations are greater than 1=2? Also nd the probability that at least one observation is less than 1/2. 2. Let X1 Binomial(n1; p) and X2 Binomial(n2; p) be independent for some n1; n2 2 Z+; p 2 [0; 1]. Let Y = X1+X2. Find the distribution of Y using one of the transform methods. 3. Using either R or Matlab plot the density of the N(; ) distribution with = 12and = 2 ??1 ??1 4 . Also draw a contour plot of this distribution (some lines of constant density in the (x1; x2) space). 4. Let MX(t) be the moment generating function (MGF) of X. Show that the MGF of Y = + X is etMX(t). Use this result to nd the MGF of Z N(; 2). By dierentiating the MGF show that EY = and VarY = 2. 5. Let X Exp(1) and (Y jX = x) Exp(2x). What is the joint pdf of X and Y ? What is the marginal pdf of Y ? 6. Using either R or Matlab use the inverse transform method to draw 200 observations from the Weibull(= 4; = 1) distribution. Show working for how you obtained the mathematical form of the transform. Then produce a plot showing all of the following: true cdf empirical cdf 95% Wald condence intervals for the true cdf based on the empirical cdf. 1

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