NAME: __________________________________________________ EMAIL:___________________________ RIN:_________________

There are eight pages with eight questions. Answer any six questions. (If you answer more, then we will grade the first ones.)

We record the order in which you hand in your exam the better to keep track of the exams, and sometimes to plot scatterplots of order vs grade. (There is never any significant correlation.) Your order does not affect your grade.

• This exam is closed book but a calculator and two 3-sided letter-paper-size note sheets are allowed.
• You may not use other computers or communication devices, or share material with other students.
• For full points, your answers must be simplified as much as reasonable.
1. [_____/10] Evaluate and give me the answer as an expression perhaps involving {$\pi$} and {$\sqrt{\cdot}$}. Show your work. {$${\cal A} = \large \int_{-\infty}^\infty e^{-x^2} dx$$}




1. [_____/10]
• Random variable X is continuous U[0,2].
• Random variable Y is continuous U[0,3].
• Random variable Z is continuous U[0,4].
• Random variable W {$\overset{\Delta}{=}$} min(X,Y,Z).
Compute P[0<W<3].

2. [_____/10] You are throwing darts uniformly at a circular dartboard with diameter 20 cm.
1. Compute the pdf {$f_{XY}(x,y)$}
2. Transform to meters: {$W \overset{\Delta}{=} X/5, Z \overset{\Delta}{=} Y/5$}. Compute the pdf {$f_{WZ}(w,z)$}.
3. Transform {$(X,Y)$} to {$(R,\Theta)$} and compute {$f_{R\Theta}(\rho,\theta)$}.

3. [_____/10]
• Random variable X is continuous U[-2, 2].
• Random variable Y is defined as {$Y \overset{\Delta}{=} X^2$}.
Compute {$f_X(x), F_X(x), E[X], VAR[X], f_Y(y), F_Y(y), E[Y], VAR[Y], COV[X,Y], \rho_{XY}$}.

4. [_____/10] You toss a fair coin 10000 times.
1. Compute the probability that the number of heads is between 5050 and 5200 inclusive using the central limit theorem. Express your answer using the Z function.
2. You observe 5100 heads from your experiment. You know that the mean number of heads is 5000.
If you repeated this experiment (of 10000 tosses) many times, what's the probability that you would see a number of heads at least this far from the mean in either direction? Express your answer using Z.

5. [_____/10] Assume that a tricycle tire's probability of failing in the next hour is 0.1 and does not depend on its age. (That is appropriate if it fails from road hazards and not from wearing thin.)
1. What's the name of the appropriate probability distribution?
2. You have 3 tires and no spares. What's the probability of neither failing in the next 4 hours?
3. Now you're stocking supplies for a team of 10 tricyclists, so 30 tires are in use at any time. They race for 10 hours each. Compute the mean and standard deviation for the number of tires you expect to wear out.

6. [_____/10] You toss 2 4-sided dice, a red one and a blue one.
• Each has faces numbered 1 to 4.
• The red one is fair.
• The blue one has P[1] = P[2] = 0.2, P[3] = P[4] = 0.3.
1. What's the probability that your total is at least 6?
2. What's the probability that your total is at least 6 given that the red die shows 3 or 4?
3. You win an amount equal to the absolute value of the difference between the two dice. What's the expected value of your win?

7. [_____/10] A signal X is transmitted over a noisy channel. X is Gaussian with mean 0 and standard deviation 5. The channel adds noise N. N is Gaussian with mean 0 and standard deviation 5. The received signal is {$Y \overset{\Delta}{=}X+N$}.
What is the maximum a posteriori value for X given Y=5.

(end of exam /WRF)