1. [15 pts] Use Definition 1.5 (definition of probability function) to prove Propo- sition 1.3 ()...
(20 pts) Write down the definition of kernel which is defined on R. Prove that the function defined, for any x,y ER, K(x,y) e-\x-yl? is a PSD kernel. Can you also figure out the feature map Ø and the feature space H associated with this kernel. = e
Problem 1. Let A event from outcome space S,equipped with probability function I . Prove that P(A) 1. Hint: You can use theorem 1.4 Problem 2. Let A,B,C events from outcome space S, equipped with probability function P. Prove that P(AUBUC)- P(A)+P(B)+P(C)- PAnB) PAnC) PBnC) +P(AnBnc) Hint: You can treat A and BUC as two events and apply theorem 1.6. You will also need to use Law 5 from the distributive laws.
Problem 1. Let A event from outcome space S,equipped with probability function I . Prove that P(A) 1. Hint: You can use theorem 1.4 Problem 2. Let A,B,C events from outcome space S, equipped with probability function P. Prove that P(AUBUC)- P(A)+P(B)+P(C)- PAnB) PAnC) PBnC) +P(AnBnc) Hint: You can treat A and BUC as two events and apply theorem 1.6. You will also need to use Law 5 from the distributive laws.
5. Suppose X ~Exp(A). (a) [5 pts] Show that E(X) 1/A. [Hint: You can directly use the definition and properties of a gamma function.] (b) [5 pts] Prove that P(x >t+ |xs) P(x > t) for s,t>0. [Hint: You can directly use the tail probability P(X > x) = e-k for x > 0.
hint This exercise 5 to use the definition of Riemann integral F. Let f : [a, b] → R be a bounded function. Suppose there exist a sequence of partitions {Pk} of [a, b] such that lim (U(Pk, f) – L (Pk,f)) = 0. k20 Show that f is Riemann integrable and that Så f = lim (U(P«, f)) = lim (L (Pk,f)). k- k0 1,0 < x <1 - Suppose f : [-1, 1] → R is defined as...
Question 1 1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function...
5. Suppose XExp(A). (a) [5 pts) Show that E(X) 1/A. Hint: You can directly use the definition and properties of a gamma function.] (b) [5 pts] Prove that P(X > t +s | X > s) = P(X > t) for s, t > 0, [Hint: You can directly use the tail probablity P(X >) e for 0.]
Question 1. A Discrete Distribution - PME Verify that p(x) is a probability mass function (pmf) and calculate the following for a random variable X with this pmf 1.25 1.5 | 1.7522.45 p(x) 0.25 0.35 0.1 0.150.15 (a) P(X S 2) (b) P(X 1.65) (c) P(X = 1.5) (d) P(X<1.3 or X 221) e) The mean (f) The variance. (g) Sketch the cumulative distribution function (edf). Note that it exhibits jumps and is a right continuous function.
exercice 6 6. The goal of this problem is to prove that a function is Riemann integrable if and only if its set of discontinuities has measure 0. So, assume f: a, bR is a bounded function. Define the oscillation of f at , w(f:z) by and for e >0 let Consider the following claims: i- Show that the limit in the definition of the oscillation always exists and that f is continuous at a if and only if w(f;...
1. (2 pts) If A and B are events, where P(B) > 0, then show that P(A' |B) 1- P(A | B) Hint: Use the definition of conditional probability to show that P(A | B) P(A' | B) 1. derived the Law of Total Probability (the simple case) Take a look at how we 4. (2 pts) At the beginning of the week, George made seven sandwiches: three turkey sandwiches two ham sandwiches, and two roast beef sandwiches. Each day,...