Question

X is said to have a uniform distribution between (0, c), denoted as X ⇠ U(0, c), if its probability density function f(x) has the following form f(x) = (1 c , if x 2 (0, c), 0 , otherwise .

(a) (2pts) Write down the pdf for X ⇠ U(0, 2).

(b) (3pts) Find the cumulative distribution function (cdf) F(x) of X ⇠ U(0, 2). (Caution: Please specify the function values for all 1

(c) Find the mean, second moment, variance, and standard deviation for X ⇠ U(0, 2).

(d) Let Y be the amount of coffee in ounces that will be dispensed when a certain automatic coffee machine is activated. Suppose that the machine never dispenses less than 6 or more than 8 ounces. Within this range, any quantity is equally likely to be dispensed. If we define X = Y 6, then X ⇠ U(0, 2). Use the result from (c) to derive the mean and variance for Y .

(e)Suppose that the coffee machine was activated for 100 times during a day. Use central limit theorem to find the probability that the total amount of co↵ee in ounces that was dispensed was less than 710 ounces.

Answer 1

Let p.d.f of X be fX(x) and c.d.f of X be FX(x)