(e) Are X and Y independent rv's? Explain. Ox and Y are independent because P(x, y) = Px(x) · Py(y). OX and Y are not independent because P(x, y) = Px(x) · Py(y). OX and Y are independent because P(x, y) OX and Y are not independent because P(x, y) Px(x) · Py(y). Px(x) ·Py(y). A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island at a particular time, and let Y denote the number of hoses on the full-service island in use at that time. The joint pmf of X and Y appears in the accompanying tabulation. p(x, y) 0 1 2 0 0.10 0.05 0.01 x 1 0.06 0.20 0.07 2 0.06 0.14 0.31 (a) What is P(X = 1 and Y = 1)? P(X = 1 and Y = 1) = | (b) Compute P(X ≤ 1 and Y < 1). P(X ≤1 and Y < 1) = (c) Give a word description of the event { X0 and Y + 0 }. One hose is in use on one island. One hose is in use on both islands. At least one hose is in use at both islands. At most one hose is in use at both islands. Compute the probability of this event. P(X +0 and Y 0) = [ (d) Compute the marginal pmf of X. x 0 Px(x) Compute the marginal pmf of Y. y Py(y) 0 Using Px(x), what is P(X ≤ 1)? P(X ≤ 1) = = 1 2 1 2

Simplify the following Boolean functions, using three-variable K-maps: F = A’B’ + BC + A’BC’

Solve both the questions!!!

1. Consider the following functions of n. (1) (2) (3) (4) (5) (6) (7) (8) (9) fi(n) = n, if n = 2.k for k = 0, 1, 2,... = n², otherwise. f2(n) = n, if n = 3.k for k= 0, 1, 2,... = n², otherwise. f3(n) = log n. Answer the following questions and prove your answ Yes or No. Yes or No. Yes or No. Yes or No. Yes or No. Yes or No. Yes or No. Yes or No. Yes or No. fi(n) = 2(f₂(n)), f2(n) = Q(fi(n)), fi(n) = 0 (f₂(n)), f₂(n) = 0 (fi(n)), fi(n) = O(n), f₂(n) = O(n²), f3(n) = O(√n), f3(n) = O(fi(n)), 22n = 0(2¹), swer for each one.

2. Show that each of the following formulas are both satisfiable and falsifiable. (a) y Vxp(x, y). (b) Vr (p(x) V q(x)) → (Vxp(x)) v (Vxq(x)) (c) Vx (yp(x,y) →yq(x, y)).

Show me the steps of determine red

Q4/ Given the Boolean function F(A, B, C, D) =(0, 1, 6, 7, 1O, 11) together with the don't care conditions d (A, B,C,D)= (2,3,4,5,8,9,14,15). a) Simplify F in sum of products (SOP). b) Simplify F in product of sums (POS). c)implement the function f in both

Compute the following values of ψ(X, B), the number of B-smooth numbers between 2 and X. (c) ψ(50, 7)

Can you represent the following Boolean function f(A, B) with a linear classifier? If yes, show the linear classifier. If not, briefly explain why not. f(A, B) A B 1 1 1 1

I'm having trouble understanding what exactly makes a linear ordinary differential equation "linear". I know that the major advantage of the linear ODEs is that a linear combination of particular solutions gives another particular solution. Characterized by additivity and homogeneity, this means that the output for a sum of inputs is equal to the sum of outputs for each individual input (e.g., f(3) + f(5) = f(3+5)= f(8)) and scaling the input by a factor scales the output by the same factor (e.g., f(6x) =6f(x)...but how would this look? Like say if I had 4e^5(t-1) satisfying the differential equation dy/dt=5y where y(1)=4, what would that "linear combination" look like?