6. Let X be a normal random variable with mean u = 10. What is the standard deviation o if it is known that p (IX – 101 <>) =
Show that this grammar is ambiguous for the string a+b+c: <S> - <x> <X> - <x>+ <x> <X> - <id> <id> - abc Give the derivations.
PROB5 Let U and V be independent r.v's such that the p.d.f of U is fu(u) = { 2 OSU< 27, otherwise. and the p.d.f'of2 is Seu, v>0, fv (v otherwise. Let X = V2V cos U and Y = 2V sin U. Show that X and Y are independent standard normal variables N(0,1).
Problem 2.13 - page 31. Let G be an n-vertex graph such that for any non-adjacent vertices U, V EV(G), d(u) + d(u) > n. Prove that G is Hamiltonian
b. Let U2 u~xã. Show that E(b)=n-2 for n > 2. LU
(5) Use induction to show that Ig(n) <n for all n > 1.
4. Let X, Y, and Z be independent random variables, each with the standard normal distribution. Compute the following: (a) P[X + Y> Z +2 (b) Var3x 4Y;
Let n ez, n > 0; let do, d1,..., dn, Co,..., En be integers in the range {0, 1, 2, 3,4}. Prove: If 5*dx = 5* ex k=0 k=0 then ek = =dfor k = 0,1,...,n.
Let X, Y be two independent exponential random variables with means 1 and 3, respectively. Find P(X> Y)
Let f(x, y) 2e-(x+y), x > 0, y > 0. Show that X, Y are independent. What are the marginal PDFS of each?