1. Let, y eR. Determine if the following are metrics on R or not (a) d, (x, y) = /1x - y| (b) d2(x,y) = (x – y)*
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
![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%2F735fc8ef-48cc-41bf-8ab8-891c98b94d58%2Fd1beux4_processed.jpeg&w=3840&q=75)
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