Most galaxies take the form of a flattened disc with the major part of the light coming from this very thin fundamental plane. The degree of flattening differs from galaxy to galaxy. In the Milky Way Galaxy, most gases are concentrated near the center of the fundamental plane. Let X denote the perpendicular distance from this center to a gaseous mass. X is normally distributed with mean 0 and standard deviation 100 parsecs. (A parsec is equal to approximately 19.2 trillion miles.) (McGraw-Hill Encyclopedia of Science and Technology, vol. 6, 1971, p. 10.)
(a) Sketch a graph of the density for X. Indicate on this graph the probability that a gaseous mass is located within 200 parsecs of the center of the fundamental plane. Find this probability.
(b) Approximately what percentage of the gaseous masses are located more than 250 parsecs from the center of the plane?
(c) What distance has the property that 20% of the gaseous masses are at least this far from the fundamental plane?
(d) What is the moment generating function for X?