3. Suppose that we observe two random variables X1, X2, each with a Poisson distribution and mean parameters di and A2 respectively (the pmf for the Poisson distribution in in section 2.4 of the notes, page 23 in backgroundIntro.pdf). For i = 1, 2, let O; = log A¡ = a + Bzi where z1 = 0 and z2 = 1. Suppose that we observe X1 = x1 and X2 = r2 (x; 2 0). (a) Compute the maximum likelihood estimate of a under the null hypothesis Ho: B = 0. (b) Derive the score test for Ho: B = 0. (c) Now let X1+X2 = m and derive the conditional distribution of X2 given m as a function of a and B, a2 and m. (d) Derive the score test for Ho: B = 0 using the conditional distribution. (e) Suppose that in a two-armed randomized trial with equal numbers of subjects per arm we observe 65 and 49 deaths in groups A and B respectively. Assuming that the numbers of deaths follow a Poisson distribution, perform the score test for the null hypothesis that the death rates are the same in the two arms. (Note that this procedure gives a "quick and dirty" uway of assessing the statistical significance of event rates between two equal sized groups.)

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.