Show that we have the analogous bound
for the case of an arbitrary, but countable, number of events
[Hint: use the limit properties of the probability function.]
Show that we have the analogous bound for the case of an arbitrary, but countable, number...
Please do only Problem 4! Use 3 as result. 3. Use the inclusion-exclusion formula derived in class as well as induction on the integer n to show that for any sequence of events {AjlI, we have that j-1 This upper bound is referred to as the union bound. 4. Extend the above result to show that we have the analogous bound P( A) P(A), j-1 for the case of an arbitrary, but countable, number of events } Hint: Use the...
Let be an arbitrary function and A X. i) Show that A ii) Give an example to show that in general A = . iii) Show that, if is injective, then A = iv) Show that, if X and Y are modules; is a homomorphism of modules and A is a submodule of X such that ker, then we also have A = We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe...
3. (8 marks) Let be the set of integers that are not divisible by 3. Prove that is a countable set by finding a bijection between the set and the set of integers , which we know is countable from class. (You need to prove that your function is a bijection.) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let be an arbitrary mapping satisfying the properties (S1) - (S4) of Theorem (at the end). Beyond that let Show that the following statements apply to all u, v ∈ Rn. The Theorem: For the scalar product, vectors u, v, w∈ Rn : We were unable to transcribe this imageWe were unable to transcribe this imageu_u We were unable to transcribe this image u_u
Suppose we could take the system of (Figure 1) and divide it into an arbitrary number of pipeline stages k, each having a delay of 300/k, and with each pipeline register having a delay of 20 ps. Figure: We were unable to transcribe this imageWe were unable to transcribe this imagePart B What would be the throughput of the system? Express your answer in terms of k. vec Throughput GIPS Part c What would be the ultimate limit on the...
a.) Is monotone? why? b.) it is bounded above by what number? Bounded below by what number? (c) Find its limit and prove it use this as hint please help, I need help on these We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Suppose that is nonempty and bounded above. Then has a supremum. Note: Show that there is a least element such that is an upper bound for . if is not a least upper bound for , show there is at least such that is an upper bound for . Proceed in this way to find the supremum. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image21 We were unable to...
Let be a sequence of independent random variables with and . Show that in probability, We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Electrodynamics. Consider a linear medium where and are both zero in the region of interest. Show that the Maxwell's equations are invariant to the transformation where is a dimensionless constant and is a constant but arbitrary angle. In other words, if and are solutions of Maxwell's equations, show that and too. Consider the special case and thus show that, in this sense, the fields and can be interchanged. This property is often named the duality property of the electromagnetic field....
Let be independent, identically distributed random variables with . Let and for , . (a) Show that is a martingale. (b) Explain why satisfies the conditions of the martingale convergence theorem (c) Let . Explain why (Hint: there are at least two ways to show this. One is to consider and use the law of large numbers. Another is to note that with probability one does not converge) (d) Use the optional sampling theorem to determine the probability that ever attains...