Problems to be turned in: 1. An experiment has four equiprobable outcomes, S (1,2,3,4). We define...
l. An experiment has four equiprobable outcomes, 'S-{ 1 ,2, 3, 4). We define three events: A-{ I , 2), В-{ 2, 3 } and C-{ l , 3). Are A, B and C independent? Justify your answer.
1 . An experiment has four equiprobable outcomes. S { 2.3) and C { 1 , 2.3.4). We define three events A { 1 . 2 }. В { 1.3). Are A. B and C independent? Justify your answer.
Problem 1.2 Consider an experiment with sample space S = {1,2,3,4}. Define events A, B, C as A = {1,2}, B = {2,3}, C = {1,4}. (a) Are A, B, C mutually disjoint? Are A, B, C collectively exhaustive? (b) Is it possible to have P[A] + P[B] + P[C] = 1? Explain why or why not. (c) If P[A] + P[B] + PIC] = 1, what is the value of P[A]?
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...
Consider the experiment of picking a four-digit PIN uniformly at random over all possible four-digit PINs, and define the random variables: X = number of distinct digits (i.e. how many different digits appear once or more) Y = length of longest streak of the same digit Below are the values of the random variables for some sample outcomes: X(1,3,1, 3)) = 2, Y((1,3,1,3)) = 1, X(2, 4, 3, 3)) = 3, Y(2, 4, 3, 3)) = 2, X (2,2, 4,...
In an experiment, there are n independent trials. For each trial, there are three outcomes, A, B, and C. For each trial, the probability of outcome A is 0.20; the probability of outcome B is 0.70; and the probability of outcome C is 0.10. Suppose there are 10 trials. (a) Can we use the binomial experiment model to determine the probability of four outcomes of type A, five of type B, and one of type C? Explain. No. A binomial...
In an experiment, there are n independent trials. For each trial, there are three outcomes, A, B, and C. For each trial, the probability of outcome A is 0.40; the probability of outcome B is 0.40; and the probability of outcome C is 0.20. Suppose there are 10 trials. Can we use the binomial experiment model to determine the probability of four outcomes of type A, five of type B, and one of type C? Explain.
We conduct an experiment where there are only four possible outcomes:A, B, C, or D. There are four possible distributions on these outcomes corresponding to θ 0, 1, 2, or 3 respectively. These distributions are A 0.25 0.5 0.120.8 B 0.250.25 0.13 0.1 C0.25 0.13 0.25 0.05 D 0.250.12 0.50.05 I want a test that decides between the null hypothesis θ = 0 versus the alternative θ in other words, the alternative that θ is either 1, 2, or 3)...
(b) Construct an experiment and three associated events A, B and C such that A and B are not independent, but AC and BC are independent. Justify your answer with calculations
7. Suppose that an experiment has two outcomes 0 or 1 (such as flipping a coin). Suppose that independent experiments and for the ith experiment you let the random variable X Ber(p) with we will assume for this problem that p is the same for each i). Then, you run n tell you the outcome for 1 isn. Then we can assume that for each i, that X p P(X 1) (where ΣΧ. let X (a) What is the state...