Solve problem 3.19
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.
Solve problem 3.19
![3.2. Order Topology.
Definition 3.15. Let X be a nondegenerate set with linear order <. The order topology on
X is the topology generated by the subbasis S consisting of all positive and negative open
rays; that is,
S = {(-∞, b) | b E X}U{(a, ∞) | a E X}.
Exercise 3.16. Let X be a nondegenerate ordered set with the order topology. Show the
following: (1) every open interval is open, (2) every closed interval is closed, (3) every open
ray is open, and (4) every closed ray is closed.
Problem 3.17. Suppose X has no least nor greatest element. Describe the basis B for the
order topology generated by the subbasis S. Show that B\ S is also a basis for the order
topology on X.
Problem 3.18. Suppose X =
basis B for the order topology generated by the subbasis S. Is B\S also
topology on X ?
[a, w] has both a least and a greatest element. Describe the
basis for the order
Problem 3.19. For X = N with the natural order <, describe the order topology on N.
Have we seen this topology on N before? Find a minimal basis for the order topology on N.
Remark 3.20. For the real numbers R with the natural order <, the basis described in
Problem 3.17 is called the standard basis for the standard topology on R. For the unit
interval (0, 1] with the natural order < (inherited from the order on R), the basis described
in Problem 3.18 is called the standard basis for the standard topology on [0, 1].
Problem 3.21. Give R the standard (order) topology. Show that every point of R is a limit
point of R. Is every point of R a sequential limit point?
Problem 3.22. Let X = [0, 1] CR. On the one hand, X has a subspace topology T, induced
by the standard (order) topology on R. On the other hand, X has a natural order induced
by the natural order on R. This order induces an order topology T2 on X. How are T1 and
T2 related?
Problem 3.23. Let X = {; | n E Z+}U {0}C R. (1) What is the relationship between
the subspace topology on X and the order topology on X? (2) Give X the order topology.
What are the limit points of X?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53e9ab80-64ca-4d50-9be9-589e5309635d%2Fbfa39f3e-a61d-471f-982e-4d144b890bbd%2Fd97jkth_processed.jpeg&w=3840&q=75)
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