Set 4 Due April 26, 2016 BENG 100 S16 1. See iPython notebook on Ted. Fill it in, run it, and make sure it works. Submit on Ted. 2. Suppose that Y is a geometric random variable of parameter p, 1 (a) show that E[Y ] = p . (b) show that P (X > k + j|X > j) = P (X > k). This is termed the memoryless property of the geometric distribution. 3. Consider a random variable X with pmf PX (k) = e k k! for some 0 and nonnegative integers k. (a) Show that the PMF indeed sums to one. (hint: think about Taylor series). (b) Express E[X] in terms of . 4. If the weather is good (which happens with probability 0.6), Alice walks the 2 miles to class at a speed of V = 5 miles per hour, and otherwise drives her motorcycle at a speed of V = 30 miles per hour. What is the mean of the time T to get to class? Hint: First derive the pmf of T and then calculate its mean. (The purpose of this problem is to illustrate that in general, E[g(X)] = g(E[X])). 5. Consider a pair of random variables (X, Y ) that are equally likely to appear an a circle of radius between 3 and 6. (a) Draw the support of their joint distribution. (b) Note that fX,Y (u, v) = C if (u, v) lies in the support and otherwise it is 0. Use part (a) and calculate C. (c) Calculate cov(X, Y ). (d) Are X and Y statistically independent? (hint: if the answer is no, you need to simply nd one pair u, v such that fX,Y (u, v) = fX (u)fY (v)). 6. Suppose that X and Y are independent Gaussians with 0 mean and variance 2 . (a) Describe their joint density fX,Y (u, v) (b) Calculate P X 2 + Y 2 a2 hint: transform to polar coordinates. (c) Let Z = X 2 + Y 2 . Calculate the density fZ (w) for any w 0 (hint: it is one of our commonly used random variables from the notes). 7. (a) For any random variable Y and a set A, dene the indicator random variable 1{Y A} to be the random variable that takes the value 1 if Y () A and is 0 otherwise. Show that E[1{Y A} ] = P (Y A). 1 Problem Set 4 Due April 26, 2016 BENG 100 S16 (b) Show that if random variables Y and Z satisfy Y () Z() for any , then the random variable W = Z Y has non-negative expectation (hint: use the denition of the expectation and some simple logical reasoning). (c) Show the Markov inequality: for any non-negative random variable Y , P (Y > c) E[Y ] . (hint: consider the scenario when Y > c and c when Y c; try to provide a simple bound in both cases; use parts a and b). (d) Now consider an arbitrary random variable X. Dene the nonnegative random variable Y as Y = (X X )2 . Show that P Y > a2 2 X . a2 (Hint: this doesn't take much eort.) (e) use the previous part to derive the Chebyshev inequality: P (|X X | > a) 2 X a2 (hint: remember equivalence of events). 8. Consider two random variables X1 and X2 with joint PMF PX1 ,X2 (j1 , j2 ). (a) Let Y = X1 + X2 . Use LOTUS and the total probability theorem to write an E[Y ] in terms of E[X1 ] and E[X2 ]. (b) Let Z = Y + X3 . Use the same argument as above to conclude that E[Z] = E[Y ] + E[X3 ]. n 1 (c) Dene Xn = n i=1 Xi . Show that if each Xi has the same mean given by , then E[Xn ] = . 9. Suppose that X1 , X2 , . . . are statistically independent random variables. (a) Show that if Y = X1 X2 , then E[Y ] = E[X1 ]E[X2 ]. (b) Dene Sn = use part a). n i=1 Xi . Show that var(S2 ) = var(X1 ) + var(X2 ) (hint: (c) Show, using part (b), that var(S3 ) = var(X1 ) + var(X2 ) + var(X3 ). What is var(Sn ) for any n? (d) Now suppose that in addition to X1 , X2 , . . . being statistically independent, they are identically distributed: each Xi has the same PMF PXi (j) PX (j). (remark: the notion of random variables being independent, identically distributed is sometimes abbreviated as IID). Dene Xn = Sn , and dene 2 to be the variance of any Xi that has n 2 PMF PX . Show carefully that var Xn = . n 2 Problem Set 4 Due April 26, 2016 BENG 100 S16 (e) Use problem 7e, 8c, and problem 9d to upper bound, for any the probability P |Xn | > . >0 What happens as n ? Argue that this means that the Xn sequence of random variables tends to a constant, . You have demonstrated the law of large numbers!. 3