3. (15%) Let X is an exponential random variable with rate λ. Let a denote a constant, a > 0, and Y = ae^X

(a) \((5\%)\) Find the cumulative density function (cdf) of Y.

(b) \((2\%)\) Find the probability density function (pdf) of Y.

(c) \((8\%)\) Find E(Y) and Var(Y)

4. (10%) China Airline quote a flight time of 2 hours, 5 minutes for its flights from Taipei to Shanghai. Suppose we believe that actual flight times are uniformly distributed between 2 hours and 2 hours, 20 minutes. (a) \((5\%)\) What is the probability will be more than 10 minutes late?

(b) \((5\%)\) What are the mean and variance of the flight times?

5. (10%) The following results come from two independent random samples taken of two different normal populations. Assume that two variances are unknown, but they are the same. | Sample 1 | Sample 2 | | --- | --- | | n=15 | m=10 | | x̄ = 13.6 | ȳ = 11.6 | | S_x = 2.2 | S_y = 3.0 | (a) \((5\%)\) What is the point estimate of the difference between the two population means?

(b) \((5\%)\) Find a \(95\%\) confidence interval for the different between the two population means.

6. (10%) The following table contains observed frequencies of a sample of 500. Test for independence of the row and column variables using α=0.01 level of significance.

7. (10%) In an experiment to investigate the performance of four different brands of tires for 125-cc motorcycles, five tires of each brand were tested and number of miles until failure was observed. A partially completed ANOVA table is given: | Source of variation | Sum of Squares | Degrees of Freedoms | Mean Square | F-value | | --- | --- | --- | --- | --- | | Treatments | (?) | (?) | (?) | (?) | | Error | 235419 | (?) | (?) | | | Total | 310501 | (?) | | | (a) \((7\%)\) Complete the ANOVA table.

(b) \((3\%)\) At the \(\alpha = 0.05\) level of significance, test whether the average number of miles until failure for the four different brands of tires are equal.

8. (20%) Given are following observations collected in a regression study on two variables. Consider the linear regression model $$Y = \beta_0 + \beta_1X + \varepsilon$$ | $$x_t$$ | 2 | 3 | 4 | 5 | 6 | | --- | --- | --- | --- | --- | --- | | $$y_t$$ | 3 | 7 | 5 | 11 | 14 | (a) \((5\%)\) Find the estimated regression equation for these data.

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