A cereal manufacturer is aware that the weight of the product in the box varies slightly from box to box. In fact, considerable historical data have allowed the determination of the density function that describes the probability structure for the weight (in ounces). Letting X be the random variable weight, in ounces, the density function can be described as f(x)=2 5, 23.75 x 26.25, 0, elsewhere. (a) Verify that this is a valid density function. (b) Determine the probability that the weight is smaller than 24 ounces. (c) The company desires that the weight exceeding 26 ounces be an extremely rare occurrence. What is the probability that this rare occurrence does actually occur?

ST 701 Week Seven Notes MaLyn Lawhorn September 26, 2017 and September 28, 2017 Quick Note From Last Class There are two different ways to calculate E[g(x)] and those are R ▯ E[g(x)] = g(x)f (x)dx X R ▯ Y = g(x); then E[Y ] = yfY(y)dy Moments th n th For special g functions, we deﬁne the nmoment of X to be E[X ], n = 1;2;▯▯▯ and n central moment E[(X ▯ E(X)) ]. Moments give us information about the distribution. Examples: ▯ 1st moment, E[X], is the mean, ▯ ▯ 2nd central moment, E[(X ▯ E[X]) ], is the variance We are usually only interested in low-or