Let s = {k=1CkXAz be a simple function, where {A1, A2, ... , An} are disjoint....
Suppose that A1,A2,.., Ak are mutually exclusive events and P(B)>0. Prove that
Solve this using Z transform table and algebraic manipulation of the function. Make it as simple as possible. 3. Z(z) = (1 + + 22); ROC: 121 > 1/2
3. (14 pts.) Let the sequence an be defined by ao = -2, a1 = 38 and an = 2an-1 + 15an-2 for all integers n > 2. Prove that for every integer n > 0, an = 4(5") + 2(-3)n+1.
5. Let S = {vi, u2, , v) be a set of k vectors in Rn with k > n. Show that S cannot be a basis for Rn.
i. (2nd Principle of Induction): Suppose that a1 = 2 and a2 = 4 and for n > 2, an = 5an-1 – 6an-2. Prove that for all n e N, an = 2". (This is easy. Show precisely where you need the 2nd Principle.)
2 Double summation Let a1, A2, A3, ... be a sequence of real numbers, and let n > 1 be an integer. Which of the following are always equal? пі пп nn nn « ££« Žia. E£« L« i=1 j=1 i=1 j=1 i= 1 i=1 j=1 j=1 i=j
Let a1 = 3 and an+1 = a + 5 2an for all n > 1 Prove that (an)nen converges and find limn7oo an:
Let z=5 where x, y, z E R. Prove that z? +z2+z?>
Find the probability that Y is greater than 3. Let Y have the probability density function f(y) = 2/y3 if y> 1, f(y) = 0 elsewhere.
Question 2 7 pts Theorem If A1, A2, .., A, are sets for n > 2, then (A, UA, U... A.) = (A) n(A)n... n(A) Upload Choose a File Question 3 6 pts o el DLL