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- Suppose that we observe that X1, X2, . . . , Xn are iid∼ U(0, 1). Show that X(1)converges in probability to zero.If ƒ is continuous on [a, b] and, what can you conclude about ƒ?Let {Xn} and {Yn} be sequences of random variables such that Xn diverges to ∞ in probability and Yn is bounded in probability. Show that Xn +Yn diverges to ∞ in probability.
- The central limit theorem tells us (select all that apply ) - that the population of a sample means an be regarded as normal no matter what the sample size is - that as long as the sample size is greater than 30, the parent population can be assumed about normal -that as long as the sample size is greater than 30, the distribution of the sample means is about normal -that no matter what the parent population looks like the distribution of the sample means can be assumed to be normalProve that the limit of xy[(x^2 - y^2)/(x^2 + y^2)] is zero as (x, y) goes to (0,0)Suppose f is uniformly continuous on [n, n + 1] ∀ n ∈ N. Does it follow that f is uniformly continuous on R?