IV. (15%) The city has recently installed red light cameras to discourage traffic signal violations. The Department of Transportation has collected the number of violations for the week prior to the installation of the camera (the first row) and the week after the installation (the second row). Based on the data, can we conclude at 5% significance level that the camera can reduce traffic light violations? Explain your answer.

1. (15%) Let X₁, X₂, X₃ denote a random sample of size n = 3 from a distribution with the p.d.f. $$f(x) = \binom{2}{3} \binom{3}{2}^{x-1}, x=1, 2, 3, \dots.$$ That is, X₁, X₂, X₃ are mutually independent and each has the geometric distribution. (a) (5%) Compute \( \mathrm{P}(\mathrm{X}1 = 1, \mathrm{X}2 = 2, \mathrm{X}_3 = 1) \).

(b) \((5\%)\) Determine \(\mathrm{P}(\mathrm{X}1 + \mathrm{X}2 + \mathrm{X}_3 = 4)\).

(c) (5%) If Y equals the maximum of \( X{1}, X{2}, X{3} \), find \( P(Y \leq 2) = P(X{1} \leq 2)P(X{2} \leq 2)P(X{3} \leq 2) \). (Please show the process of calculation and round to four decimal places)

2. (15%) Let X₁, X₂, ..., Xₙ be a random sample of size n from uniform distribution on interval (θ-1, θ+1). (a) (5%) Find the method of moments estimator for \(\theta\).

(b) \((5\%)\) Is your estimator in part (a) an unbiased estimator for \(\theta\)

(c) (5%) Given the following \( n = 5 \) observations of \( X \), give a point estimate of \( \theta \): 6.6, 7.7, 6.9, 8.3, 7.5. (Please show the process of calculation)

3. (10%) A random variable X has a Poisson distribution with λ > 0. Given X, a random variable Y is N(μ, X). (a) (5%) What is the E(XY)? (b) (5%) What is the \(\operatorname{Var}(\mathbf{Y})\)? (HINT: \(\operatorname{Var}(\mathbf{Y}) = \operatorname{E}(\mathbf{Y}^2) - \operatorname{E}(\mathbf{Y})^2\) (Please show the process of calculation)

4. \((10\%)\) \(X{1}, X{2}, \ldots, X{n}\) are the independent and identically distributed random variables and follow the distribution with mean \(\mu\) and variance \(\sigma^2\). Let \(\bar{X} = \sum{i=0}^{n} X_i / n\). Please show that \(\bar{X}\) is a consistency estimator of \(\mu\).

5. (10%) A test of epidemic disease has the following results: to those patients who have been infected, the correct rate (showing positive) is \(90\%\); however, to the healthy people, there still exists \(1\%\) error rate (showing positive). Assume the prevalence rate is \(0.1\%\) (that is, there are 10 persons really infected in 10000 sample). If a man's test reports positive, what is the probability he really infected?

6. (10%) A random variable \( X \) follows lognormal distribution, \( \log X \sim N(\mu, \sigma^2) \). Derive the probability density function of \( X \).

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