a.? ?Show that if ?X ?has a normal distribution with parameters ? and ? , then Y = aX + b (a linear function of ?X?) also has a normal distribution. What are the parameters of the distribution of ?Y ?[i.e., ?E?(?Y?) and ?V?(?Y?)]? [?Hint?: Write the cdf of Y?,P(Y ?y) , as an integral involving the pdf of ?X?, and then differentiate with respect to ?y? o get the pdf of ?Y.? ] b.? ?If, when measured in °C , temperature is normally distributed with mean 115 and standard deviation 2, what can be said about the distribution of temperature measured in ° F?

Answer: Step1 of 2: a). To show that if X has a normal distribution with parameters and . Then the linear function of x is Y = aX + b this is also a normal distribution. Here we need to find the parameters of the distribution Y. Then, x follows a normal distribution with mean and variance . That is, X ~ N(,) The pdf of ‘x’ is (x) f xx) = 1 e 2 ; - < x < 2 Differentiate on both sides We get, 1 (x) f xx)dx = e 2 dx ……..(1) 2 Given the linear function is Y = aX + b Y - b = aX yb x = a yb et x = a = g(y) Substitute the ‘x’ value in equation (1). yb 2 1 ( a ) f xg(y))dg(y) = e 22 dg(y) 2 2 1 ( (yb2a) dy fx(g(y))dg(y) = e 2 a .a 2 2 ((yy )) 1 22 fy(y) d(y) = e y dy ( here, y= a +xb 2a y= a )x 2 ( (yy ) 1 2 y So, fy(y) = 2a e Therefore, y is also a normal distribution with meana + banx variance a.