4. (2 pts) The domain for the variables x, y are integers. Let us be given...
2. Let Q(x,y) be the statement "x - 2 = 5y" and let the domain for x and y be all integers. Determine the truth value of each of the following statements. Justify your answers shortly. (i) Q(-3,-1) (ii) Vx3yQ(x,y) (iii) 3xVy-Q(x,y)
Let the P(x, y) be the statement x = y + 1 and assume the domain consists of all real numbers (Explain it pls) 1. What is the truth value of ∃x∀yP(x, y)? 2. What is the truth value of ∀y∃xP(x, y)?
8. (10 pts.) Find a common domain for the variables x, y, and z for which the statement is true and another domain for which it is false. Explain your answer.
Let the domain for x and y be R, the set of real numbers. (a) Determine the truth value of ∀x∃y (y = √ x). Explain (b) Determine the truth value of ∃y∀x (y = √ x). Explain
5. Let the joint cumulative density function of random variables X and Y be given by for z 0, y >0. (Note: Fxy(x, y)-0 outside this domain.) (a) Find P(X S2,Y (b) Find P(X5). (c) Find P(2 <Y s6). (d) Find the joint probability density function f(x, y). Show that your answer satisfies the S 2). two defining properties of a density. (e) Are X and Y independent? Why or why not?
1. Fundamentals: (a) Briefly, state why probability is important for statisticians (b) Let random variables X, Y, and Z be distributed according to the following table. probability 1/4 1/4 i. True or false: X and Y are independent. Explain. ii. True or false: X and Y are conditionally independent given Z. Explain. (c) Let A, B, and D be events, where 0< PD) 1. i. Prove that P(An B P(AB) 2 P(A) +P(B) 1. ii. Suppose that P(AD) 2 P(B|D)...
Let X and Y be jointly continuous random variables with joint probability density given by f(x, y) = 12/5(2x − x2 − xy) for 0 < x < 1, 0 < y < 1 0 otherwise (a) Find the marginal densities for X and Y . (b) Find the conditional density for X given Y = y and the conditional density for Y given X = x. (c) Compute the probability P(1/2 < X < 1|Y =1/4). (d) Determine whether...
Let X and Y be jointly continuous random variables with joint probability density given by f(x, y) = 12/5(2x − x2 − xy) for 0 < x < 1, 0 < y < 1 0 otherwise (a) Find the marginal densities for X and Y . (b) Find the conditional density for X given Y = y and the conditional density for Y given X = x. (c) Compute the probability P(1/2 < X < 1|Y =1/4). (d) Determine whether...
true and false propositions with quantifiers. Answer the following questions in the space provided below. 1. For each proposition below, first determine its truth value, then negate the proposition and simplify (using De Morgan's laws) to eliminate all – symbols. All variables are from the domain of integers. (a) 3.0, x2 <. (b) Vr, ((x2 = 0) + (0 = 0)). (c) 3. Vy (2 > 0) (y >0 <y)). 2. Consider the predicates defined below. Take the domain to...
4. (25 pts, 25/6 pts each) Let X and Y be random variables of the continuous type having the joint p.d.f. f(x, y) = 8xy,0 £ x £ y £ 1. 1) Draw a graph that illustrates the domain of this p.d.f. 2) Calculate the marginal p.d.f.s of X and Y. 3) Compute 4) Compute 5) Write out the equation of the least squares regression line and draw it in a graph. 6) If your calculations are correct, in 3)...