The expected value of a random variable is a weighted average of the values of a random variable.

Variance, which tells us how spread out the values of this random variable are

The expected value, also called the expectation or mean, of the random variable X on the sample space S is equal to

**E(X) = ∑**_{ s ∈ S}p(s)X(s)The deviation of X at s ∈ S is

**X(s) − E(X)**, the difference between the value of X and the mean of XThe expected number of successes when "n" mutually independent Bernoulli trials are performed, where "p" is the probability of success on each trial, is

**np**.

**Q. Let X be the number that comes up when a fair die is rolled. What
is the expected value of X?**

The random variable X takes the values 1, 2, 3, 4, 5, or 6, each with probability 1/6. It follows that

E(X) = 1/ 6 · 1 + 1/ 6 · 2 + 1/ 6 · 3 + 1/ 6 · 4 + 1/ 6 · 5 + 1/ 6 · 6 = 21/ 6 = 7/ 2 .

**Q. A fair coin is ﬂipped three times. Let S be the sample space of the eight possible outcomes, and
let X be the random variable that assigns to an outcome the number of heads in this outcome.
What is the expected value of X?**

The values of X are the eight possible outcomes when a coin is ﬂipped three times

Because the coin is fair and the ﬂips are independent, the probability of each outcome is 1/8

E(X) = 1/ 8 [X(HHH) + X(HHT) + X(HTH) + X(THH) + X(TTH) +X(THT) + X(HTT) + X(TTT)]

= 1 / 8 (3 + 2 + 2 + 2 + 1 + 1 + 1 + 0)

= 12 / 8

= 3 / 2

**Q. What is the expected value of the sum of the numbers that appear when a pair of fair dice is
rolled?**

Let X be the random variable equal to the sum of the numbers that appear when a pair of dice is rolled. The value of X for the 36 outcomes of this experiment.

The range of X is {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.

p(X = 2) = p(X = 12) = 1/36,

p(X = 3) = p(X = 11) = 2/36 = 1/18,

p(X = 4) = p(X = 10) = 3/36 = 1/12,

p(X = 5) = p(X = 9) = 4/36 = 1/9

p(X = 6) = p(X = 8) = 5/36

p(X = 7) = 6/36 = 1/6

E(X) = 2 * 1/36 + 3 * 1/18 + 4 * 1/12 + 5 * 1/9 + 6 * 5/ 36 + 7 * 1/6 + 8 * 5/36 + 9 * 1/9 + 10 * 1/12 + 11 * 1/18 + 12 * 1/36

= 7

Let A and B be sets. A binary relation from A to B is a subset of

**A × B**.A

**binary**relation from "A" to "B" is a set "R" of ordered pairs where the ﬁrst element of each ordered pair comes from "A" and the second element comes from "B".The notation

**a R b**is used to denote that (a, b) ∈ R. Moreover, when (a, b) belongs to R, "a" is said to be related to "b" by RBinary relations represent relationships between the elements of two sets.

**Example of binary relations :**

Let "A" be the set of students in your school, and let "B" be the set of courses. Let "R" be the relation that consists of those pairs (a, b), where "a" is a student enrolled in course "b".

So assume Sanjay dutt is also enrolled in CS510, then the pair (Sanjay dutt, CS510) is in R.

However, if Sharukh Khan is not enrolled in CS510, then the pair (Sharukh Khan, CS510) is not in R.

Note that if a student is not currently enrolled in any courses there will be no pairs in "R" that have this student as the

**ﬁrst element**. Similarly, if a course is not currently being offered there will be no pairs in "R" that have this course as their**second element**.

**Q. Let A ={0, 1, 2} and B ={a, b}. Then {(0, a), (0, b), (1, a), (2,b)} is a relation from A to B.
This means, for instance, that 0 R a, but that 1 R b. Relations can be represented graphically,
as shown in Figure below, using arrows to represent ordered pairs. Another way to represent this
relation is to use a table, which is also done in Figure below. **

A

**relation**on a set A is a relation from A to A.

**Q. Let A be the set {1, 2, 3, 4}. Which ordered pairs are in the relation R ={(a, b) | a divides b}?**

Because (a, b) is in R if and only if a and b are positive integers not exceeding 4 such that a divides b, we see that

R ={(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)}.

**Q. Consider these relations on the set of integers:
R _{1} = {(a, b) | a ≤ b}, R _{2} = {(a, b) | a > b},
R _{3} = {(a, b) | a = b or a = −b}, R _{4} = {(a, b) | a = b}, R _{4} = {(a, b) | a = b + 1}, R _{5} = {(a, b) | a + b ≤ 3}. Which of these relations contain each of the pairs (1, 1), (1, 2), (2, 1), (1, −1), and (2, 2)?**

The pair (1, 1) is in R

_{1}, R_{3}, R_{4}and R_{6}; (1, 2) is in R_{1}and R_{6}; (2, 1) is in R_{2}, R_{5}and R_{6}(1, −1) is in R

_{2}, R_{3}, and R_{6}; andﬁnally, (2, 2) is in R

_{1}, R_{3}and R_{4}.

**Q. How many relations are there on a set with n elements?**

A relation on a set "A" is a subset of

**A × A**.Because A × A has n

^{2}elements when "A" has "n" elements, and a set with "m" elements has 2^{m}subsets, there are 2^{n2}subsets of**A × A**.Thus, there are

**2**relations on a set with "n" elements.^{n2}For example, there are

**2**relations on the set {a, b, c}.^{32}= 2^{9}= 512

**Reflexive relations :**A relation R on a set A is called reﬂexive if (a, a) ∈ R for every element a ∈ A.**Symmetric and Anti-Symmetric relations :**A relation R on a set "A" is called symmetric if**(b, a) ∈ R**whenever (a, b) ∈ R, for all (a, b ∈ A). A relation R on a set A such that for all a, b ∈ A, if (a, b) ∈ R and (b, a) ∈ R, then a = b is called**anti-symmetric**.**Transitive relations :**A relation R on a set A is called transitive if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A.

**Q. Consider the following relations on {1, 2, 3, 4}:
R _{1}
={(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)},
R_{
2}
={(1, 1), (1, 2), (2, 1)},
R_{
3}
={(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)},
R_{
4}
={(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)},
R_{
5}
={(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)},
R_{
6}
={(3, 4)}.
Which of these relations are reﬂexive?**

The relations R

_{ 3}and R_{5}are reﬂexive because they both contain all pairs of the form (a, a), namely, (1, 1), (2, 2), (3, 3), and (4, 4).The other relations are not reﬂexive because they do not contain all of these ordered pairs. In particular, R

_{1}, R_{2}, R_{4}, and R_{6}are not reﬂexive because (3, 3) is not in any of these relations.

**Q. Is the “divides” relation on the set of positive integers reﬂexive?**

Because a | a whenever a is a positive integer, the “divides” relation is reﬂexive.

(Note that if we replace the set of positive integers with the set of all integers the relation is not reﬂexive because by deﬁnition 0 does not divide 0.)

In some relations an element is related to a second element if and only if the second element is also related to the ﬁrst element. The relation consisting of pairs (x, y), where x and y are students at your school with at least one common class has this property.

**Q. How many reﬂexive relations are there on a set with n elements?**

A relation R on a set A is a subset of A × A.

Consequently, a relation is determined by specifying whether each of the n 2 ordered pairs in A × A is in R

However, if R is reﬂexive, each of the n ordered pairs (a, a) for a ∈ A must be in R.

Each of the other n(n − 1) ordered pairs of the form (a, b), where a ≠ b, may or may not be in R.

Hence, by the product rule for counting, there are

**2**reﬂexive relations^{n(n−1)}

**Q. R1 ={(a, b) | a ≤ b},
R2 ={(a, b) | a > b},
R3 ={(a, b) | a = b or a = −b},
R4 ={(a, b) | a = b},
R5 ={(a, b) | a = b + 1},
R6 ={(a, b) | a + b ≤ 3}. Which of these relations are symmetric and which are antisymmetric?**

R3 is symmetric as if a = b or a = -b then b = a or b = -a.

R4 is symmetric as a = b implies that b = a

R6 is symmetric because a + b ≤ 3 implies that b + a ≤ 3.

R1 is anti-symmetric as the inequalities a ≤ b and b ≤ a imply that

**a = b**R2 is anti-symmetric because it is possible that a > b and b > a

R4 is antisymmetric, because two elements are related with respect to R4 if and only if they are equal.

R5 is antisymmetric because it is impossible that a = b + 1 and b = a + 1.

**Q. Consider the following relations on {1, 2, 3, 4}:
R _{1}
={(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)},
,
R_{
4}
={(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}.
Which of these relations are transitive?**

R4 is transitive, because (3, 2) and (2, 1), (4, 2) and (2, 1), (4, 3) and (3, 1), and (4, 3) and (3, 2) are the only such sets of pairs, and (3, 1), (4, 1), and (4, 2) belong to R4

R1 is not transitive because (3, 4) and (4, 1) belong to R1 , but(3, 1) does not.

**Q. Is the “divides” relation on the set of positive integers transitive?**

Suppose that a divides b and b divides c.

Then there are positive integers k and l such that b = ak and c = bl. Hence, c = a(kl),soa divides c

It follows that this relation is transitive.

Let A = {1, 2, 3} and B = {1, 2, 3, 4}. The relations R

_{1}= {(1, 1), (2, 2), (3, 3)} and R_{2}={(1, 1), (1, 2), (1, 3), (1, 4)} can be combined to obtain :-R

_{1}∪ R_{2}= {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (3, 3)}R

_{1}∩ R_{2}= {(1, 1)}R

_{1}- R_{2}= {(2, 2), (3, 3)}R

_{2}- R_{1}= {(1, 2), (1, 3), (1, 4)}

**Q. Let A and B be the set of all students and the set of all courses at a school, respectively.
Suppose that R _{1}
consists of all ordered pairs (a, b), where "a" is a student who has taken course "b",
and R_{2}
consists of all ordered pairs (a, b), where "a" is a student who requires course "b" to graduate.
What are the relations R_{1} ∩ R_{2}, R_{1} ∪ R_{2}, R_{1} ⊕ R_{2}, R_{1} - R_{2} and R_{2} - R_{1} ?**

**R**consists of all ordered pairs (a, b), where a is a student who either has taken course b or needs course b to graduate._{1}∪ R_{2}**R**is the set of all ordered pairs (a, b), where a is a student who has taken course b and needs this course to graduate._{1}∩ R_{2}**R**consists of all ordered pairs (a, b), where student a has taken course b but does not need it to graduate or needs course b to graduate but has not taken it._{1}⊕ R_{2}**R**is the set of ordered pairs (a, b), where a has taken course b but does not need it to graduate; that is, b is an elective course that a has taken._{1}- R_{2}**R**is the set of all ordered pairs (a, b), where b is a course that a needs to graduate but has not taken._{2}- R_{1}

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