2. Let S be the sample space of a single toss of a fair coin. Define the sequence of random varia...
Stats questions list sample space find odds etc Toss a fair coin 3 times, and observe the sequence of heads and tails. a. List the sample space. Let event A 2 H and 1 T, event B (At least 1 H, event C (H on the second toss) Find: b. P(A) C. P (B) d. P (C) f. P(A UC) How many ways can an executive committee of 3 be chosen from a committee of 15? ) How many ways...
Problem 1. A biased coin with probability plandin with a Heads is lipped 4 times. (a) Define the basic random variables and give the sample space and assign probabilities to the outcomes. (b) Let X be the total number of Heads in the four flips Draw a Venn diagrain showing the five events X = ii 0,1,2,3,4 as well as the sample space and the outcomes. Is X a random variable? c) Are the events X 1 and X 2...
5. (15 pts) Let S denote the sample space of tossing the HK dollar coin 9 times with success probability pon the Number side and failure probability g = 1-pon the Flower side. For i=1,2,..., 100, let X, denote the random variable on 2, having value 1 for the outcomes w i th in the number sicle and zero otherwise. Let Y = 3.X1 +3.X2 + ... +3X100- (a)(2 pts) Are the random variables X1,..., X, independent? (b)(3 pts) Find...
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following convergence results independently (i.e, do not conclude the weaker convergence modes from the stronger ones). d Yn 0. a. P b.Y 0. L 0, for all r 1 Yn C. a.s d. Y 0. Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following...
I need help with A, B, and C Toss a fair coin 8 times. Let X be the number of heads. (a) Find P(X= 3). (b) Find P(X 25). (c) Find P(X 56). (a) P(X = 3) = (Round to four decimal places as needed.)
3. (a) (5 points) Let Xi,... be a sequence of independent identically distributed random variables e of tnduqendent idente onm the interval (o, 1] and let Compute the (almost surely) limit of Yn (b) (5 points) Let X1, X2,... be independent randon variables such that Xn is a discrete random variable uniform on the set {1, 2, . . . , n + 1]. Let Yn = min(X1,X2, . . . , Xn} be the smallest value among Xj,Xn. Show...
2. SUPPLEMENTAL QUESTION 1 (a) Toss a fair coin so that with probability pheads occurs and with probability p tails occurs. Let X be the number of heads and Y be the number of tails. Prove X and Y are dependent (b) Now, toss the same coin n times, where n is a random integer with Poisson distribution: n~Poisson(A) Let X be the random variable counting the number of heads, Y the random variable counting the number of tails. Prove...
A fair coin is tossed twice. Let X and Y be random variables such that: -X = 1 if the first toss is heads, and X = 0 otherwise. -Y = 1 if both tosses are heads, and Y = 0 otherwise. Determine whether or not X and Y are independent. So far, I have determined the the joint probability distribution as follows: x = 0 x = 1 y = 0 2/4 1/4 y = 1 0 1/4
A, B, and C please Toss a fair coin 11 times. Let X be the number of heads. (a) Find P(X= 7). (b) Find P(X29). (c) Find P(X 58). (a) P(X = 7) = (Round to four decimal places as needed.)
2. (8 Marks - 2 points each) Suppose you toss a fair coin four times. Let the random variable X be the number of tails (T) obtained. Also, let E and V denote the mean and variance respectively. a) Compute E(X) and V(X). b) Compute E(5X - 6) and V (6 +5X).