Y ~ Uni (0,3), still assume event A as {Y ≤ 1}, event B as {Y ≤ 2]. Are they independent this time?
Let X, Y, Z be random variables. Prove or disprove the following statements. (That means, you need to either write down a formal proof, or give a counterexample.) (a) If X and Y are (unconditionally) independent, is it true that X and Y are conditionally indepen- dent given Z? (b) If X and Y are conditionally independent given Z, is it true that X and Y are (unconditionally) independent?
Assume that event A occurs with probability 0.6 and event B does not occur with probability 0.6. Assume that A and B are disjoint events. The probability that both events occur (A and B) is a) 0 b)0.3 c) 1.0 d) 0.7
Event A occurs with probability 0.3, and event B occurs with probability 0.4. If A and B are independent, we may concludeA. P(A and B) = 0.12.B. P(A|B) = 0.3.C. P(B|A) = 0.4.D. All of the above
Proble m 3. Let T: V ->W be (1) Prove that if T is then T(),... ,T(Fm)} is a linearly indepen dent subset of W (2) Prove that if the image of any linearly in depen dent subset of V is linearly indepen dent then T is injective (3) Suppose that {,... ,b,b^1,...,5} is Prove that T(b1), .. . , T(b,)} is a basis of im(T) (4) Let v1,. Vk} be T(v1),..,T(vk) span W lin ear transform ation between vector...
Let exp(-т*) + vk Yk where dent M and V N(0, o2 are mutually indepen R, k = 1, (a) Construct the likelihood T(y|x) and the negative log-likelihood. (b) Compute the maximum likelihood estimate îML (c) Bonus question: How does the estimate change if E(k) t0?
Let exp(-т*) + vk Yk where dent M and V N(0, o2 are mutually indepen R, k = 1, (a) Construct the likelihood T(y|x) and the negative log-likelihood. (b) Compute the maximum likelihood estimate...
Assume a composite made from a fibre volume fraction of 50% using uni directional, continuous fibres and a thermoset matrix. Sketch one stress-strain diagram showing the expected tensile behaviour of the fibre, matrix and composite, loaded in a direction parallel with the fibre orientation Use the following values for the properties of the matrix and fibre: Strain to failure (% Strength (GPa) Fibre 2 Matrix 0.3 10
Assume a composite made from a fibre volume fraction of 50% using uni...
The probability of event A is P(A) = 0.5 and the probability of event B is P(B) = 0.3. (Express all answers as decimals; do not include unnecessary decimal places--i.e. answers should be in the form 0.2 or .2, and NOT 0.20, 2/10 or 20%.) a) Find P(A and B) if A and B are disjoint. b) Find P(A or B) if A and B are disjoint. c) Find P(A or B) if P(A and B) = 0.2. d) Find...
Question 2. Consider the following data: data 1 2.9 3.7 1.0 2.4 5.1 0.3 0.4 data 2 9.0 1.8 4.5 4.8 7.9 9.9 6.8 2.2 3.5 (a). Neither of the two datasets are bell-shaped. Assume the two datasets are random, indepen dent samples. Which test should we use to compare the two datasets? Why? (b). Use the test from the previous part to determine if the first distribution is shifted to the left of the second distribution.
3.21. Problem. (Section 11.2) In each of the following cases below, assume that X and Y are independent random variables then use the Convolution Theorem to derive the proba- bility density function of X +Y. (a) The random variable X is uniform distribution on 0, 1) and the random variable Y is an exponential distribution with = 0.2. (b) The random variable X is a uniform distribution on (0, 2) and the random variable Y is a uniform distribution on...
Let A be symmetric, Y N(O,V), and w...,ws be indepen- dent X2(1) random variables. Show that for some value of sand some numbers λί Hint: YQZ so Y'AY~Z'OAQZ. Write Q'AQ PD()P