Question 3 [10 marks]. Suppose that X, Y and Z are statistically independent random variables, each of them with a x²(2) distribution. (a) Find the moment generating function of U = X + 3Y + Z. State clearly and justify all steps taken. (b) Calculate the expectation E(U) using the moment generating function.

Please could you explain why 0.5 was added to each upper limpit of the intervals.Thanks

Open Middle Problem #1: You can only use the digits 1 through 9 one time each in the seven boxes below. How can you arrange the digits to make the fraction multiplication problem true? 8.8.08 =

21/solve the following differential equation Using laplace transform y₁ =-y, Y₁(0)=1 y' = Y Y₂(0)=0 2 21 Solve the following equations: dy 1- dt + 2y + = ydt 2 cost, y(0) = 1 2 2- y(t) = ±² + (yet) sin (t-u) du Q3: Answer the following: 1- L [Log Sa] (5²+9²) 2- L1 [Log (Cos²y1] 0-25 3-L-1 [ -] 4- (5+1) 3 L (ezt sin3t) e-s 5- L-1 ( 너 (0) 5² +5

LESSON MATHEMATICS ACTIVITIES 1.3 DECIMALS 1. Josh used itres of ster during an Integrahed Express the ami remained DATE b) 14.07 2. Express 0.5 as a fraction in its simplest form or e) 327.034 7. Write the number form is e) 5.11x 10 ese standard 3. Express 0.145 os simplest form fraction in its b) 2.03x102 4. In August 2022, the cost of a litre of petrol was Sh 159.25. How much did kipchoge pay for two litres? c) 6.3x103 5. A doctor prescribed 12.5 ml of a dr to a patient. Express the drug prescribed in litres to two significant figures 8. Work out: a) 2.05 5.2-1.8 rite each of the following in stand- ds form 0039 b)3.6 2.8 (2.8+0

RK 119 43 Previous Problem University at Buffalo Problem List Next Problem Match the surfaces (a) - (f) below with the contour diagrams (1) - (6) below those. (a) Surface (a) matches contour 5 V V (b) Surface (b) matches contour 2 V (c) Surface (c) matches contour 1 (d) Surface (d) matches contour 6 V (e) Surface (e) matches contour 4 V (f) Surface (f) matches contour 3 V (4) (1) -0.25 (a) (b) (c) (d) (e) y y 2.5 0.5 1.5 1.5 1.5 y .3 0.25 OC 0.25 -0.25 (2) X 1.5 ZI (f) y 0.01 0.01 (3) ☑ X 0.01 0.01 0.2 0.2 (5) 0 x 0.5 (6) 0.25 X X 0.25 0.5

43 University at Buffalo Previous Problem Problem List Next Problem At least one of the answers above is NOT correct. The figure shows a hill with two paths, A and B. (a) What is the elevation change along each path? 400 9400 ✓ feet (b) Which path ascends more rapidly? A v (c) On which path will you probably have a better view of the surrounding countryside (assuming that trees do not block your view)? A V (d) Along which path is there more likely to be a stream? A V Note: You can earn 50% partial credit for 2-3 correct answers. Preview My Answers Submit Answers Q hulu )))) 9800' A 10000 (Click on graph to enlarge) L ^ B 0 Logged in as Luella Ya 4)

28. (a) Under what conditions do we say that two random variables X and Y are independent? (b) Demonstrate that if X and Y are independent, then it follows that E(XY) = E(X)E(Y); (e) Show by a counter example that the converse of (ii) is not necessarily true.

7. [10 marks] Let G = (V,E) be a 3-connected graph with at least 6 vertices. Let C be a cycle in G of length 5. We show how to find a longer cycle in G. (a) Let x be a vertex of G that is not on C. Show that there are three C-paths Po, P1, P2 that are disjoint except at the shared initial vertex and only intersect C at their final vertices. (b) Show that at least two of P0, P1, P2 have final vertices that are adjacent along C. (c) Combine two of Po, P1, P2 with C to produce a cycle in G that is longer than C.