MATH 8A - INTRODUCTION TO PROBABILITY PRACTICE MIDTERM # FALL 8 Name (Last, First): Student ID: TA: SO AS TO NOT DISTURB OTHER STUDENTS, EVERY- ONE MUST STAY UNTIL THE EXAM IS COMPLETE. ANSWERS TO THE TRUE/FALSE QUESTIONS DO NOT NEED TO BE JUSTIFIED; HOWEVER, INCORRECT AN- SWERS TO THE TRUE/FALSE QUESTIONS ARE PE- NALIZED. IN PARTICULAR, A CORRECT ANSWER IS WORTH 5 POINTS, AN INCORRECT ANSWER IS WORTH -5 POINTS, AND A BLANK ANSWER IS WORTH POINTS. REMEMBER THIS EXAM IS GRADED BY A HUMAN BEING. WRITE YOUR SOLUTIONS NEATLY AND CO- HERENTLY, OR THEY RISK NOT RECEIVING FULL CREDIT. THIS EXAM WILL BE SCANNED. MAKE SURE YOU WRITE ALL SOLUTIONS IN THE SPACES PROVIDED. THE ACTUAL MIDTERM EXAM CONSISTS OF 5 TRUE/FALSE QUESTIONS AND 3 LONGER FORMAT QUESTIONS. YOUR ANSWERS TO THE LONGER FOR- MAT QUESTIONS SHOULD BE CAREFULLY JUSTI- FIED. YOU ARE ALLOWED TO USE RESULTS FROM THE TEXTBOOK, HOMEWORK, AND LECTURE.
FALL 8. (5 points) Label the following statements as true or false. Any ambiguous answer (for example, resembling a hybrid of T and F) will be treated as an incorrect answer. (a) F The median of a continuous random variable is unique. (b) F If X Exp(λ) and s, t >, then P(X < s X < s + t) P(X < t). (c) T If X N (µ, σ ), then E[X ] σ + µ. (d) F If a random variable is not discrete, then it is continuous. (e) T If X is a continuous random variable and F X is its CDF, then for s, t R such that s < t, we have the equality F X (t) F X (s) P(s < X < t). (f) T If X Exp(λ) and Y Exp(η) are independent, then min(x, Y ) Exp(λ + η).
MATH 8A - INTRODUCTION TO PROBABILITY PRACTICE MIDTERM # 3. (5 points) You are trying to get to a party, which is 3 minutes away by bus. Suppose that the waiting time for a bus is distributed according to an exponential distribution with a rate of two buses every hour. If you wait longer than 3 minutes for a bus, you decide to give up and take a taxi to the party. The taxi ride to the party only takes 5 minutes. Let Y be the amount of time it takes for you to get to the party, which includes BOTH the wait time and the travel time. (a) (5 pts) Calculate the CDF of Y. You may choose to write Y in terms of hours or in terms of minutes. Proof. We work in terms of hours. Let X Exp() model the waiting time for the bus. The statement of the problem implies that Y is the following function of X: X + Y f(x) + 4 3 4 if X if X >. Make sure you understand why this is the case. Now, we can simply compute the CDF of Y : F Y (t) P(Y t) P(f(X) t) ({ P X + t, X } { 3 4 t, X > }) P ({X + } { t X }) ({ } 3 { + P 4 t X > }). We can compute these probabilities using the multiplication rule. For the first term, this means P ({X + } { t X }) ( P X ) ( P X + t X ). Since X Exp(), ( P X ) ( ) F X e x dx e. On the exam, you are allowed to simply cite the values of the CDF of the exponential distribution without computing the integral. Of course, there is little partial credit to be had for simply citing an answer that s wrong, so make sure you get it right on your cheat sheet. You could also do the actual calculation by hand, but this seems like an unwise strategy considering the time constraint of the midterm.
4 FALL 8 Now, note that ( P X + t X ) ( P X t if t < ({ } { }) X ) P X t X [ ] ( ) if t P X, if t > if t < ( ) P X t [ ] ( ) if t P X, if t > if t < ) F X (t ( ) [ ] e t e F if t, X if t > We are not done yet as there is one more term to compute, but this one is much easier now that we have written it properly, namely, ( ({ } 3 { P 4 t X > }) P X > ) e if t 3 4 if t < 3 4. Putting all of our work together, we arrive at the CDF if t < ( e ) [ e t e F Y (t) if t, 3 ) 4 ( e ) [ ] e t 3 e + e if t 4, if t >.
MATH 8A - INTRODUCTION TO PROBABILITY PRACTICE MIDTERM # 5 (b) ( pts) Calculate the expected value E[Y ]. Again, you may choose to write Y in terms of hours or in terms of minutes. Proof. Again, we compute everything in terms of hours. The calculation follows easily from the fact that E[Y ] E[f(X)] and X Exp(). In particular, note that Y is not a continuous random variable; however X is continuous with a simple density. So, E[Y ] E[f(X)] ( x + ) ( e x ) f(x)(e x ) dx ( + ) ( e ( ) ) e + + e ( ) + e x dx + 3 4 ( + e + e + + 3 4 e 3 4e. ( x + ) ( e () ) + e () + 3 4 e( )( ) ) (e x ) dx + e x dx ( e x ) + 3 ( 4 P X > ) ( ) 3 (e x ) dx 4 The algebra on the actual midterm will certainly not be any messier than this; however, you should prepare yourself adequately.
6 FALL 8 3. Let X and Y be independent continuous random variables with CDFs F X and F Y respectively. We also use the notation f X and f Y to denote their respective densities. { X if X Y (a) ( pts) Compute the density of the random variable Z min(x, Y ) Y if X > Y in terms of the functions F X, F Y, f X, f Y. Proof. Note that F Z (r) P(Z r) P(Z > r) P(X > r and Y > r) P(X > r)p(y > r) ( F X (r))( F Y (r)). This implies that F Z is continuous; moreover, this implies that F Z is differentiable at all but finitely many points. In particular, f Z (r) F Z(r) d [ ] ( F X (r))( F Y (r)) d [ ] ( F X (r))( F Y (r)) dr dr ( ) ( F X (r))( f Y (r)) + ( f X (r))( F Y (r)).
MATH 8A - INTRODUCTION TO PROBABILITY PRACTICE MIDTERM # 7 (b) (5 pts) Compute the density of the random variable Z X + Y. Your answer should be in terms of the densities f X and f Y of X and Y respectively. Proof. Note that P(a Z b) P(a X + Y b) Thus, the density f Z (y) of Z is f Z (y) a x+y b a x y b x b x a x b b a a f X (x)f Y (y) dydx f X (x)f Y (y) dydx f X (x)f Y (y) dydx f X (x)f Y (y x) dydx ( ) f X (x)f Y (y x) dx dy. f X (x)f Y (y x) dx.
8 FALL 8 4. Let P (X, Y ) be a uniform random point in the rectangle R [, ] [, ] {(x, y) : x, y }. (a) ( pts) What is the joint density f (X,Y ) (x, y) of P? What is the marginal density f X (x) of X? What is the marginal density f Y (y) of Y? Are X and Y independent? Proof. Since P is uniform on the rectangle R, the joint density is simply equal to rectangle R. Formally, if x and y, f (X,Y ) (x, y) otherwise. Area(R) in the Of course, X and Y. The marginal density can be computed by integrating out the other coordinate: f f X (x) (X,Y ) (x, y) dy dy if x, otherwise; and f f X (x) (X,Y ) (x, y) dx dy if y, otherwise. Since f (X,Y ) (x, y) f X (x)f Y (y), we conclude that X and Y are independent.
MATH 8A - INTRODUCTION TO PROBABILITY PRACTICE MIDTERM # 9 (b) (5 pts) Let Z X + Y be the sum of the coordinates of the random point P. Determine the CDF of Z. Compute the expectation E[Z]. Proof. One can use 3(b) to compute this density; however, a simple picture will be a lot easier to work with (and captures the same idea). In particular, F Z (r) P(Z r) P(X + Y r). Draw the line y r x. The region inside of the rectangle R AND below the line y r x correspond to the outcomes in the event (Z r). For r [, ], this region looks like a triangle and the area is simply r r. For r (, ], this region looks like a trapezoid with area. For r (, 3], this region looks our rectangle but with a triangle cut off from the top right corner: in particular, this region has area (3 r). Putting this altogether, we get if r < r r 4 F Z (r) r r 4 if r [, ] if r (, ] (3 r) (3 r) if r [, 3] 4 if r > 3. Note that F Z (r) is continuous; moreover, F Z is differentiable at all but finitely many points. So, Z has a density r if r (, ), if r (, ), f Z (r) 3 r if r (, 3), otherwise. We can now compute the expectation E[Z] 3 rf Z (r) dr r r dr + r 3 dr + r 3 r dr r3 6 + r ( ) 3r 4 + 4 r3 3 6 6 + 3 4 + 5 4 9 6 3.
FALL 8 5. Suppose that the joint density of the random variables X, Y is given by the function e (x y) y if x R and y >, f (X,Y ) (x, y) π otherwise. Find the marginal density of X. Your answer should be in terms of the CDF Φ of the standard normal distribution. Proof. This amounts to computing the integral f X (x) f (X,Y ) (x, y) dy e (x y) y dy π π e (y x) y dy. Note that the integrand looks almost like the density of N (, x) except for the additional factor of y in the exponent. So, let us try to complete the square in the exponent: (y x) y y xy + x + y So, f X (x) e x e x (y (x )) f (X,Y ) (x, y) dy x y (x )y + x (y (x )) (x ) + x (x ) x (y (x )) + x π e (y (x )) dy. π e y dy e (y x) y π e x P(Z x) where Z N (, ) e x ( P(Z < x)) e x ( Φ( x)). π e (y (x )) + x dy
MATH 8A - INTRODUCTION TO PROBABILITY PRACTICE MIDTERM # (ADDITIONAL SPACE FOR WORK, CLEARLY INDICATE THE PROBLEM YOU ARE WORKING ON)
FALL 8 (ADDITIONAL SPACE FOR WORK, CLEARLY INDICATE THE PROBLEM YOU ARE WORKING ON)