Proposition 1.1 Suppose that X1, X2,... are random variables. The following quantities are random variables: (a) max{X1, X2) and min(X1, X2); (b) sup, Xn and inf, Xn; (c) lim sup∞ X and lim inf∞ Xn- (d) If Xn(w) converges for (almost) every w as n→ ∞, then lim- random variable. → Xn is a

1. Let (2, F, P) be a probability space with a filtration F = (Fn)n20. (a) Let 71, 72 be two F-stopping times. Prove that T1 A 72 = min(T1, T2) T1 VT2 = max(T1, T2) are both stopping times. (b) Let 7 be an F-stopping time. Prove that 7+1 is also an F-stopping time.

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A random walk (RW) {Sn}n>o is a sum of id r.v.s X1, X2, •• , Xn,withP(X1 = a) = p, P(X1 = b) = 1 -p= q.Find the expectation E(Snl and variance var(Sn) of Sn for any n.

3. Let X be a continuous random variable on the interval [0, 1] having cdf Fx(x) = 3x² − 2x³. Let W = 4 + 3X. (a) What is the range of W? (b) Find a formula for Fw(w). (c) Find a formula for fw(w).

2. Let U1, U2,... be independent random variables, each with continuous distribution that is uniform in the interval [-3, 3]. Let S, = U ++ Un. (a) Compute the moment generating function of U1. (b) Compute the moment generating function of S,. (c) Determine E(S) and E(S) (by any method).

Please solve (a) & (b) only!

Suppose X1, X2, ... , Xn is a random sample and Xi = {1, with probability p         0, with probability 1-p} for every i = 1, 2, ... , n.   Find the Moment Generating Function of ∑i=1n Xi . What is the distribution of ∑i=1n Xi ?

9- A college professor never finishes his lecture before the end of the hour and always finishes his lectures within 2 min after the hour. Let X = the time that elapses between the end of the hour and the end of the lecture and suppose the pdf of X is i) f(x) = { = { kd² 0 the value of k and draw th Total kx² 0≤x≤2 otherwise LE COR v) What is the expected value of X? vi) what is the variance of X? curve. [Hint: anilll Ul UlIC - Jocture continues for at 1