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3. Suppose that the incidence of accidental bleach poisoning, requiring hospitalization, inNova Scotia is believed to follow a Poisson distribution with a mean of 5.6 cases every 10years. IfXis the random variable representing the number of cases in any given year,give the following;(a) What is the probability mass function ofX?(1)(b) What isP(X= 1)?(2)(c) What isP(X≥1).(2)(d) What isμXand what isσX?(3)
4SECTION II4. Suppose it is claimed that a random variable,X, has the following distribution:f(x) =(0.09375(4-x2)-2x20Else.(a) Show this is a legitimate probability distribution.(3)(b) Find the cumulative distribution function,F(x), for the random variableX.(3)(c) ComputeP(-0.5X <1.5).(3)
55. Suppose we have the following distribution:f(x) =(314(x2+y2)1x2,1y20Else.(a) Find the marginal probability distributionsfx(x) andfy(y).(3)(b) Determine the expected values,E(X), andE(Y).(3)(c) Using the short cut formula, find the variances,V AR(X) andV AR(Y). .(3)(d) Determine the covariance,COV(X, Y).(3)(e) Find⇢x,y, .(3)
6SECTION III1. Suppose that a random variableXhas an exponential distribution.(a) What ism, the dimension of the parameter vector✓?(1)(b) Provide the Maximum Likelihood estimator,ˆ✓MLE.(5)2. Suppose that a random variableXhas a Poisson distribution.(a) What ism, the dimension of the parameter vector✓?(1)(b) What is the first population moment?(2)(c) What is the first sample moment?(2)(d) Provide the Method of Moments estimator,ˆ✓MOM.(2)
7Now suppose we have some data drawn from this Poisson distribution.