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EBK FIRST COURSE IN PROBABILITY, A
- Prove the conjecture made in the previous exercise.arrow_forwardLet R be the set of all infinite sequences of real numbers, with the operations u+v=(u1,u2,u3,......)+(v1,v2,v3,......)=(u1+v1,u2+v2,u3+v3,.....) and cu=c(u1,u2,u3,......)=(cu1,cu2,cu3,......). Determine whether R is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.arrow_forwardLet f(x),g(x),h(x)F[x] where f(x) and g(x) are relatively prime. If h(x)f(x), prove that h(x) and g(x) are relatively prime.arrow_forward
- 4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .arrow_forwardFor an element x of an ordered integral domain D, the absolute value | x | is defined by | x |={ xifx0xif0x Prove that | x |=| x | for all xD. Prove that | x |x| x | for all xD. Prove that | xy |=| x || y | for all x,yD. Prove that | x+y || x |+| y | for all x,yD. Prove that | | x || y | || xy | for all x,yD.arrow_forwardFind all monic irreducible polynomials of degree 2 over Z3.arrow_forward
- Let X and Y be independent variables taking values in the positive integers such that n (7) ₁² (1 - py²-k pk k for some p and all 0≤ k ≤n. Show that X and Y have Poisson distributions. P(X = k | X + Y = n) =arrow_forwardLet A = {9,10,11,12,13} and let f : A→N be defined by f (n) = the highest primefactor of n. Find the range of f.arrow_forwardDefine the function fn : [0, 1]→ R for n > 1, defined as if æ E (!, 1] nQ if x E (!, 1]\Q if æ € [0, 1) nQ if z € [0, 표)\Q if x: 1 4 %3D = (x)" n a) Prove that fn is a simple measurable function. b) Determine fo.1] fndju. c) Do we have fn(x) Riemann integrable on [0, 1]? d) Determine So.11 lim fndµ.arrow_forward
- Let X be a random variable on a closed and bounded interval [a, b] whose CDF is continuous. Let g(x) be a convex function. Prove that g (E (X)) ≤ E(g(X))arrow_forwardLet X - N(2,4). Calculate P[X? – 4X < 4].arrow_forwardLet 2 be a measurable set of finite measure and let 1arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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