3. Let K,and K, be compact subsets of a metric space X. Show K, U K2 is compact
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![1. Let , y ER. Determine if the following are metrics on R or not
(a) d, (x, y) = V[x – y|
(b) d2(x,y) = (x – y)*
2. Let E CX where X is a metric space. Let E° denote all the interior points of E. (E°is an open
set-you don't have to prove this, but you can use this fact to prove the following)
(a) Show E is open if and only iff E = E°
(b) Show: If GCE and G is open, then G C E°
3. Let K,and K2 be compact subsets of a metric space X. Show K, U K2 is compact
Show NaEa Ka is closed
4. Let {Ka}aea be a collection of compact subsets of a metric space,
in X
5. (a) Consider the collection of open sets, {(-n,n)}"=1 in R with the usual metric d(x,y) =
|x – y| . Use this collection of open sets to show R is not compact (make sure to prove R =
Unej(-n, n) as part of your work).
(b) Give an example of a collection of compact sets in R , say {Kn}n=1 such that U=1 Kn is
not compact (Hint: consider part (a) )
6. Let E C X where X is a metric space. Show the limit points of E are the same as the limit
points of Ē , the closure of E. That is, show E'= (E U E')',
7. (a) Let A C X,B C X where X is a metric space. Show AnB CÃ O Ē
(b) Consider A = [3,7),B = (7,9] as subset of R with the usual metric, dr (x, y) = | x – y| .
Show Ā n B is not a subset of An B
8. Let X be a metric space. Let A and B be separated sets in X. Show either E CA or ESB
where E is a connected subset of A U B](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcfc5bdab-6654-444e-b9d5-5fc4b8470fbb%2F9007f168-56ae-436b-9289-cf0e50c98243%2Fg65hk5g_processed.jpeg&w=3840&q=75)
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