1. A fast food manager wants to be prepared to have enough of COIN change (pennies, nickels, dimes, and quarters) at the drive through window ready to give to customers. Let X be the amount of COIN change that any customer is given back when they pay for their food at the drive-up window (in decimal form.) Suppose f(x) is as given below.

f(x) =kx(1-x) where 0 x < 1.00and k is some constant > 0

1. Find the value of k that makes this a legitimate pdf. (HINT: For the 2^nd condition, set the integral equal to one, do the integration, and solve for k. WHY?)

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1. Graph f(x), including a scale for the X and Y axes.

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1. Show that f(x) is a legitimate pdf for X.

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1. Find the MEAN amount of change a customer receives back when paying for their order.

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1. Using your graph from part (b) and the fact that this distribution is symmetric, what is the median of this distribution?Recall that to the left of the median, there is an area of .5, and to the right of the median, there is an area of .5.(No integration needed here due to symmetry.)

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1. Set up but do not calculate how we would find the variance of X.

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1. Then show how we take that result to find the standard deviation (use proper units.)