Question 9 Let X be an Exponential random variable, with the moment generating function (MGF): M(t) = (1-0)-¹. (a) Derive the MGF above. (b) Derive E(X) using the MGF. (c) Derive V(X) using the MGF. (d) Find the MGF of Y, where Y = 2X - 3.

5-2 Compute the derivative ficts of the logistic sigmoid 1 fax= H 50 1 + exp(-x) 0.

A child inflates a balloon, admires it for a while and then lets the air out at a constant rate. If V(t) gives the volume of the balloon at time t, then the figure shows V'(t) as a function of t. At what time does the child do each of the following? y'(t) 1 -2 3 (a) begin to inflate the balloon? t = (b) finish inflating the balloon? t = (c) begin to let the air out? it = 1 6 0-2 3 3 (d) What would the graph of V'(t) look like if the child had alternated between pinching and releasing the open end of the balloon, instead of letting the air out at a constant rate? '(t) V'(t) 6 9 6 9 12 15 v'(t) 9 12 15 18 12 t 18 t nna: 15 18 1 -2 3 6 6 9 v'(t) 9 12 15 18 12 18 t t