Statistics and probability

(c). For the log-normal random variable X = e , where Y ~ Normal(0, 62), prove that the probability density of X is 1 (In(x) - 0)2 f(x | 0, 0?) = 2 02 , 0 0, the density function is f(x | ro, 0) = 0x8x-0-1, r > ro, and 0 > 1. Note that the cumulative distribution function of X is P(X 0, define a statistic T3(X1 , . . . , Xn, Y1, . . ., Ym) which has an F distribution. An F distribution is determined by the numerator and denominator degrees of freedom. State the degrees of freedom for your statistic 13. (c). For your answer in (b), define the statistic TA(X1, . .., Xn, Yl, . .., Ym) = T3(X1 , . . . , Xn, Yl, . .., Ym) What is the distribution of Ta under the conditions of (b)? (d). Suppose that of = ?. If SX = ,, EL,(X, - X)?, and Sy = Em, (Yi-Y)?, are the sample variances of the two samples, show how to use the F distribution to find P(Sx2/ $3 > c). (e). Repeat question (d) if it is known that of = 202