IB Maths HL, SL, Studies: March 2013

Sunday, March 31, 2013

IB Maths HL - Normal Distribution

How can we find the mean $\mu$
of the weight of a population of students which is found to be normally distributed with standard deviation 2 Kg and the 30% of the students weigh at least 53 Kg.

The answer is from www.ibmaths4u.com

Let the random variable W denote the weight of the students, so that

$W\sim N( \mu, 2 ^2)$

We know that $P(W \geq 53)=0.3$

Since we don’t know the mean, we cannot use the inverse normal. Therefore we have to transform the random variable $W$ to that of

$Z\sim N(0,1)$ , using the transformation $Z= \frac{X- \mu}{\sigma}$

we have the following

$P(W \geq 53)=0.3 \Rightarrow P(\frac{X- \mu}{\sigma} \geq \frac{53- \mu}{2})=0.3$

$\Rightarrow P(Z \geq \frac{53- \mu}{2})=0.3$

Using GDC Casio fx-9860G SD
MAIN MENU > STAT>DIST(F5)>NORM(F1)>InvN>

Setting Tail: right
Area: 0.1
$\sigma$ :1
$\mu$ :0

We find that the standardized value is 0.5244

Therefore
$\frac{53- \mu}{2}=0.5244 \Rightarrow 53- \mu =1.0488$

$\mu=53-1.0488=51.9 (3 s.f.)$

Saturday, March 30, 2013

On line Integrator

IB Maths On line Integrator from wolfram.com

http://integrals.wolfram.com/index.jsp

Friday, March 29, 2013

Group Theory

A group

is a finite or infinite set of elements together with a binary operation (called the group operation) that together satisfy the four fundamental properties of closure, associativity, the identity property, and the inverse property. The operation with respect to which a group is defined is often called the "group operation," and a set is said to be a group "under" this operation. Elements

, ... with binary operation between

and

denoted

form a group if

1. Closure: If

and

are two elements in

, then the product

is also in

2. Associativity: The defined multiplication is associative, i.e., for all

3. Identity: There is an identity element

(a.k.a. 1,

, or

) such that

for every element

4. Inverse: There must be an inverse (a.k.a. reciprocal) of each element. Therefore, for each element

, the set contains an element

such that

A group is a monoid each of whose elements is invertible.

A group must contain at least one element, with the unique (up to isomorphism) single-element group known as the trivial group.

The study of groups is known as group theory. If there are a finite number of elements, the group is called a finite group and the number of elements is called thegroup order of the group. A subset of a group that is closed under the group operation and the inverse operation is called a subgroup. Subgroups are also groups, and many commonly encountered groups are in fact special subgroups of some more general larger group.

A basic example of a finite group is the symmetric group

, which is the group of permutations (or "under permutation") of

objects. The simplest infinite group is the set of integers under usual addition. For continuous groups, one can consider the real numbers or the set of

invertible matrices. These last two are examples of Lie groups.

http://mathworld.wolfram.com/Group.html

IB Maths HL Option Groups

The fundamental theorems and propositions on Group Theory from www.ibmaths4u.com

Re: IB Maths HL Option: Sets, Relations and Groups

1. $\mathbb{Z}_{n} =0,1,…,n-1$ is a group under addition modulo n. With identity 0 and the inverse of $i$ is the $n-i$

2. The number of elements of a group is its order $|G|$

3. The order of an element g, which is denoted by $|g|$ , in a group G is the smallest positive integer n such that $g^n=e$

4. Cyclic group $G=\{g^n |n\in \mathbb{Z}\}= \langle g \rangle$ and g is called a generator of $G$

5. The order of the generator is the same as the order of the group it generated.

6. Let G be a group, and let x be any element of G. Then, $\langle x \rangle$ is a subgroup of G.

7. Let $G=\langle g \rangle$ be a cyclic group of order n.
Then $G=\langle g^k \rangle$ if and only if gcd(n,k)=1.

8. Every subgroup of a cyclic group is cyclic.

9. Every cyclic group is Abelian.

10. If G is isomorphic to H then G is Abelian if and only if H is Abelian.

11. If G is isomorphic to H then G is Cyclic if and only if H is Cyclic.

12. Lagrange’s Theorem: If G is a finite group and H is a subgroup of G, then the order of H divides G.

13. In a finite group, the order of each element of the group divides the prder of the group.

14. A Group of prime order is cyclic.

15. For any $x\in G, x^{|G|}=e$ where e is the identity element of the group G.

16. An infinite cyclic group is isomorphic to the additive group $\mathbb{Z}$

17. A cyclic group of order n is isomorphic to the additive group $\mathbb{Z}_{n}$ of integers modulo n.

18. Let p be a prime. Up to isomorphism, there is exactly one group of order p.

19. Let (G, *) be a group and $x\in G$ . If $x*x=x$ , then $x=e$ , where $e$ is the identity element of the group G.

20. In a Cayley table for a group (G, *), each element appears exactly once in each row and exactly once in each column.

21. A group (G,*) is called a finite group if G has only a finite number of elements. The order of the group is the number of its elements.

22. A group with an infinite number of elements is called an infinite group.

23. If (G,*) is a group, then ({e},*) and (G,*) are subgroups of (G,*) and are called trivial.

24. Let G be a group and H be a non-empty subset of G. then H is a subgroup of G if and only if for all $a,b \in H, ab^{-1} \in H}$ .

25. Let G be a group and H be a finite non-empty subset of G. then H is a subgroup of G if and only if for all $a,b \in H, ab^{-1} \in H , ab \in H}$ .

26. Let $\langle g \rangle$ be a finite group of order n.
Then $\langle g \rangle =\{e,g,g^2,…,g^{n-1}\}$

27. Let $\langle g \rangle$ be a finite cyclic group. Then the order of g equals the order of the group.

28. A finite group G is a cyclic group if and only if there exists an element $x \in G$ such that the order of this element equals the order of the group ( $|G|$ ).

29. Let G be a finite cyclic group of order n. Then froe every positive divisor d of n, there exists a unique subgroup of G of order d.

30. Let G be a group of finite order n. Then the order of any element x of G divides n and $x^n=e$

31.Let G be a group of prime order. Then g is cyclic.