Let SL(2, R) be the set of all 2 × 2 matrices a b such that a, b, c, dɛR and ad = be = 1. Prove that SL(2, R) is a group under matrix multiplication. It is called the special linear group.
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.
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Is Ga group under the operation of function composition?
19. Do the nonzero real numbers form a group under the operation given by a * b=
Ja|b, where | a|is the absolute value of a?
CourseSart
20. Prove that S, has order n!. [Hint: There are n possible images for 1; after one
has been chosen, there are n -
1 possible images for 2; etc.]
21. Suppose G is a group with operation +. Define a new operation # on G by
a # b = b * a. Prove that G is a group under #.
22. List the elements of the group D; (the symmetries of a regular pentagon).
[Hint: The group has order 10.]
23. Let SL(2, R) be the set of all 2 X 2 matrices
a b
such that a, b, c, deR
c d
and ad = be = 1. Prove that SL(2, R) is a group under matrix multiplication.
It is called the special linear group.
24. Prove that the set of nonzero real numbers is a group under the operation *
defined by
Sab
if a > 0
a * b =
la/b if a < 0.
25. Prove that R* × R is a group under the operation + defined by (a, b) * (c, d) =
(ac, bc + d).
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