# Question 2 [12 marks Let X be the maximum temperature in the month of January in Melbourne and let Y

###### Question:

Question 2 [12 marks Let X be the maximum temperature in the month of January in Melbourne and let Y be the maximum temperature in the same month in Sydney. Historical records show that the distribution of X and Y have probability density functions fx(x), fy(y), respectively, shown in Figure 1 (a) By eye-balling the plot, explain why the population mean and variances of the maximum temperatures in Melbourne and Sydney are Hx 25°C, Hy28°C σ -1°C, and σ-2.25°C (b) Assume now that you have 10 independent measurements of maximum temperature X1o, and 5 independent measurements of maximum temp in Melbourne Λι, er ature in Sydney Yi,... . Ys (i) What are the distributions of TO 25) andY -28)? (ii) What is the probability that the sample average of the measured temperatures in Melbourne, denoted as X, will lie between 24.37°C and 25.32°C? (iii) What is the distribution of Y-X, where Υ is the sample average of the measured temperatures in Sydney? (iv) What critical temperature T, will yield PO-x T) 0.05? (X-25) and s30-28), where 陨and S루 and Y, respectively? (v) What are the distributions of the sample variances of X

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