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6/7/2013 11:16 PM

Probability laws

Complementary event

The complementary even A’ of an event A is defined as

Addition Law

Note: (1) is added at the end.

Conditional probability

|

Note: (1) | | |.

Independence

Definition: Two events A and are independent of each other if and only if Theorem: Two events A and are independent of each other if and only if | | Mutually exclusive

Theorem: Two events A and are mutually exclusive of each other if and only if Note: (1) Two events A and are mutually exclusive of each other if and only if

Law of total probability

Theorem: If are mutually exclusive and exhaustive, then | | | Bayes' theorem

Theorem: Let A1, A2, ... , An be a set of mutually exclusive events that together form the sample space S.

Let B be any event from the same sample space, such that P(B) > 0. Then,| ∑




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Probability distribution

The probability distribution describes the probability of different events. It can be represented by a list, a graph,

a table of a mathematical formula.

Probability mass function

Definition: The probability mass function (discrete probability function) is a function that allocates

probabilities to all the distinct values that the discrete random variable can take. It

can be written as a table or a function.

Note: (1) When a p.m.f. is represented by a graph, dots are present instead of continuous lines.

Probability density function

Definition: The probability density function of a continuous random variable is a function that allocates

the probabilities to all the ranges of values that the random variable can take.

Note: (1)

(2)

(3)

(4)

(5)

Distribution function

Definition: The distribution function (cumulative distribution function) is defined as

Note: (1) For a discrete random variable,

(2) For a continuous random variable,

(3) In the case of discrete random variable, F(t) is a step-function




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Expectation (or mean)

Definition: Given a random variable X with a probability distribution f(x), the expectation (or mean) of X,

denoted by E(X), is defined by

,for discrete X

,for continuous X

Note: (1) If ,then f i/N

(2) A game is fair if E(X)=0

(3) If y=ax+b, E(Y)=aE(X)+b

Population mean

Sample mean

for a discrete frequency distribution taking values with corresponding frequency , the

mean is

Note: (1) In general, Mean deviation

∑ | |

Population variance

Sample variance

Note: (1) Sample variance is defined like this such that it is an unbiased estimator for population

variance.




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Variance , -

It can be expanded as If the random variable X is continuous with probability density function p(x),

where

and where the integrals are definite integrals taken for x ranging over the range of X.

Properties for linear transformation of random variables

The variance of a finite sum of uncorrelated random variables is equal to the sum of their variances. This stems

from the identity:

and that for uncorrelated variables covariance is zero.




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Combinations and permutations

Permutation of all objects

Definition: The different groups in which a given collection of objects can be arranged by varying their

order in every possible way are called the permutation or arrangements of the objects takenall at a time.

Theorem: The number of permutations of n distinct objects which are all taken is denoted

by Theorem: The number of permutations of n objects of which p1 are alike of one kind, p2 are alike of one

kind, p3 are alike of one kind, …, is given by

Permutation of some objects from the population

Theorem: The number of permutations of n distinct objects taken r at a time is denoted by

Theorem: The total number of permutations is the individual permutation in different cases that is

Corollary: The total number of permutations of n different objects, taken not more than r at a time,

when anyone of the n objects may be repeated is given by

Combination

Definition: A combination is the arrangement of objects without reference to the order of the

arrangement in the group.

Theorem: The number of combinations of n distinct objects taken r at a time is denoted by

./ Note: (1) ./ . /

Theorem: The total number of combinations is the individual permutation in different cases that is




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Binomial distribution

Bernoulli distribution

Definition: An independent repetition of an experiment which has just two possible outcomes is called a

Bernoulli trial.

Theorem: The Bernoulli probability function f(x) of a Bernoulli random variable X is given by

Note: (1) f(x) is a function that can represent the probability of successful and not successful events in

just one expression.

(2) p must be a constant for each Bernoulli distribution.

(3) The Bernoulli random variable attains only two values 0 and 1 and is a random variable since

Theorem: The mean and variance of X are given by Binomial distribution

Theorem: The Binomial probability function f(x) of a Binomial random variable X is given by

./ Note: (1) The probability distribution of X is called is Binomial distribution with parameters n and p,

and is denoted by

(2) The experiment consists of n independent Bernoulli trials.

(3) f(x) is the probability of obtaining exactly x success in n independent Bernoulli trials.

(4) The mean and variance of X are given by

(tricky proof, do it in reverse direction)




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Normal distribution

Normal distribution

If the random variable X is in , then the mean and variance of X are given by Note: (1) The bell-shaped curve of the normal distribution is symmetric about .

(2) The flatness of a normal curve is determined by the value of .

(3) Z is in a standard normal distribution if Z is in N(0,1) with p.d.f. f(z).

(4) P(Z>0)=P(Z<0)=1/2

Theorem: Let the random variable in

.

If , then Z is in N(0,1)

Note: (1) P(X<x)=P(Z< )

(2) Approximation from normal table: Normal approximation to Binomial distribution

Theorem: If X is a Binomial variable with parameters n and p, then for large n and p not too small

nor large, then

X is approximately in N(np, np(1-p))

Note: (1) Since X is discrete, when it is to be approximated by a normal distribution which is

continuous, a continuity correction is needed.

Linear combinations of independent normal variables

Theorem: If X and Y are two independent normal variables such that X is in and Y is in , then

There isn’t any combination of Binomial variables as the Binomial distribution always refers to one sample.




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Statistical inference

Sampling distribution

Theorem: If

*

+is a random sample of size n from

, then the statistic

is in

NOTE: (1) This distribution of is called the sampling distribution of means with √ as the

standard error of the mean

Sample proportion???

Point estimation???

Pooled estimates

抄 2-7 THM 4 (1)(2)

Interval estimation

Definition: A B% confidence interval (a,b) is defined such that , which is the

probability that the unknown lies in this interval is B/100.

NOTE: (1) The construction of the confidence interval is based on the sample values of unbiased

estimator for predicting the value of an unknown parameter.

(2) The population must be normal and must be known

(3) When is unknown with n large enough, the sample variance can be used as an

estimator for .

Confidence interval for the sample mean

( )

√ √ √

NOTE: (1) The confidence interval enable us to locate an unknown population men with a certain

probability.

(2) To find the C.I. for with higher probability, the interval becomes longer. To shorten theconfidence interval, we become less confident about the location of .




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C.I. for sample proportion???

Hypothesis testing

Two-end test

Procedure:

(1)

(2)

(3)

√

(4) Choose such that ( ) , where is the level of significance

(5) || 圖

(6) The decision rule is

“Reject if √ √ One-end test

Take one example to illustrate,

(1)

(2)

(3)

√

(4) Choose such that

(5)

(6) The decision rule is

“Reject H_0 if √ ”

2-17 remarks 5.5圖

NOTE: (1) |

(2) The choice of z is subjective

(3) If z lies in the critical region, is rejected. If z lies in the acceptance region, is not

rejected

(4)

√

圖: one-end vs two-end




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Differential equations

Differential equations

First order•Variable separable

• 333333esults

• Linear DE

• -----------------

• exact DE

•First order homogeneous DE

• -----------------

Second order

•Second order linear DE which is homogeneous

•Second order linear DE which is non-homogeneous




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1. Ordinary differential equations in first order

1.1. Variable separable

A variable separable first order DE is in the form:

Then the solution is

NOTE: (1) Don’t forget the arbitrary constant.

1.2. Linear DE

A first order linear DE is in the form:

∫

∫ ∫

(∫ ) ∫

∫ ∫

∫ [ ∫ ]

If the right-hand integral is a definite one, then

- ∫ ∫

The Bernoulli’s equation

where




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1.3. Solution of exact DE in first order

The DE: is exact (result of product rule)

The solution is in the form

1.4. First order homogeneous DE

ƒ is homogeneous iff

The DE:

is homogeneous

iff

1. Use the substitution

2. Transform

NOTE: (1) If Then use the substitution where




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2. Ordinary differential equations in second order

2.1. ~

Form:

Solution:

[ ] 2.2. ~

Form: Solution: multiplying both sides by 2y’ yields

,- ,- , -

,-

which is a variable separable DE

2.3. Second order linear DE which is homogeneous

A second order linear DE which is homogeneous is in the form:

The equation is called the characteristic equation (auxiliary equation)

Case (1) The characteristic equation has two distinct real roots

The general solution is

Case (2) The characteristic equation has a repeated root

The general solution is

Case (3) The characteristic equation has a pair of complex root




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By using Euler’s formula in Case(1), the general solution is 2.4. Second order linear DE which is non-homogeneous with constant coefficients

Given the non-homogeneous equation

Let be the particular solution which depends on f(x)

The solution of the homogeneous DE is called the complementary solution.

The general solution of the non-homogeneous equation is

Case (1) When f(x) is a polynomial

(a) If 0 is not a root of the characteristic equation.

(b)

If 0 is a root of the characteristic equation.

Case(2) When

(a)

If t is not a root the characteristic equation, or t is equal to the real part of the

complex root the characteristic equation.

(b)

If t is one of the distinct real roots of the characteristic equation.

(c)

If t is the double root of the characteristic equation.

Case(3) When

(a) If

does not contain

.

(b)

If contains .

Case(4) When f(x) is a linear combination of a polynomial, , , the particular

solution of each form is form separately. Then

NOTE: (1) In case (1b), if the characteristic equation has 0 as a root, then

superposition??




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If only, after differentiation, .

(2) In case (2b), if t is a root and only, after differentiation, LHS=0.

2.5. Reducible second order DE

Case (1)

Case(2) Use the substitution suggested in the problem.

Also use the following two formulae

miss picture of results??




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Numerical method

IMPORTANT

NOTE To show that an answer is in correct to at least n decimal places, it is equivalent to prove

||

Numerical method

Interpolation

(reclaiming the shape of a unknow function)

• Lagrange Interpolating Polynomial

• Given a set of (xi, f(xi)) , interpolate from these results

•Taylor’s Expansion

• Reclaim the shape of a function about a certain point

• uses derivatives

• “Substitution”

• let p(x)=anxn+ an-1xn-1+…+0

Approximating an integral

•Trapezoidal rule• Simpson’s Rule

•Other Integration

• (invented by the author of the exam paper)

Getting a numerical solution

•Fixed-point iteration

•Newton’s Method

•Secant method

• absorbed into Method of False Position

•The method of false position




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1. Lagrange Interpolating Polynomial (L.I.P.)

2.6. first degree L.I.P p1(x) (straight line)

Given (x0, f(x0)) and (x1, f(x1))

By two points form:

After arranging terms, we shall get:

2.7. secondary degree L.I.P p2(x) (quadratic polynomial)

Given (x0, f(x0)), (x1, f(x1)) and (x2, f(x2))

NOTE This formula frequently appear in HKALE. Candidates better acquaint themselves with this formula.

2.8. nth degree L.I.P.

Given (n+1) points : (xi, f(xi)) for i=0,1,2,…,n

Let The required nth degree L.I.P.

NOTE for the interpolation of third degree polynomial, it is advised to let and

solve for the coefficients. Questions about interpolation of 4th degree L.I.P do not frequently appear in HKALE.

2.9. error of Lagrange Interpolating Polynomial

The error at any value , - using p(x) as an approximation to f(x) :

where

NOTE Rigorously, the interval for should be open as the formula is derived with mean value theorem.

is not frequently required to be solved. Instead, the bounds for E(t) or |E(t)| is frequently required.

depends on the choice of x, ie. the is distinct in E(1) and E(2).

graph




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In L.I.P., it is required to prove

| | || Condition:

i. Limits for

is given, or

ii. f(x) is given, to be differentiate by candidates

To prove p(x)=q(x)+(x-x1) (x-x2)…(x-xn)r(x) for some ,-

It is advised to show (x-x1) (x-x2)…(x-xn) is a factor of p(x)-q(x)

Usually p(x) is the new polynomial to be found; q(x) is the approximated polynomial; q(x)=f(x) and (x-x1)

(x-x2)…(x-xn)=0 at x1, x2,…xn

2. Taylor’s Expansion (T.E.)

2.10. Taylor’s Expansion for f(x) about the point x=a up to xn is in the form:

2.11. error of Taylor’s Expansion

where lies between the opened interval between a and x is the remainder term(or error term) used to estimate(compensate) the terms after the xn term which

are not included.

NOTE In some exam questions, it is defined that Candidates are required to prove

, namely. While p(x) is complicated, working out

and

is tedious.

However and which are simpler can replace p’(a) and p’’(a) when finding Taylor’s Expansion forp(x) about x=a.

ABSOLUTE CAUTION: There should be constraints for f(x) and g(x), say ,- ,-

the remainder estimation theorem

A proof of Taylor’s Theorem is on Thomas Calculus P.818

graph




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3. Trapezoidal Rule

2.12. ordinary Trapezoidal Rule

(proved by integrating an L.I.P.)

∫ , -

2.13. error of ordinary Trapezoidal Rule

2.14. composite Trapezoidal Rule

(obtained from sum of results of ordinary Trapezoidal Rule)

, -

n = no. of sub-intervals

≠no. of points given

NOTE (n+1) points are given.

2.15. error of composite Trapezoidal Rule

NOTE When the curve concave upwards, the result is over-estimated and vice versa.

A proof is on Thomas Calculus P.606

4. Simpson’s Rule

2.16. ordinary Simpson’s Rule

, -

NOTE The above formula is better to be memorized as Simpson’s Rule for more than 3 points is not frequently

required.

graph

graph




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2.17. error of ordinary Simpson’s Rule

where ,-

2.18. composite Simpson’s Rule

, -

“odd” points “even points”

NOTE There are (2N+1) points and 2N subintervals. A quadratic curve is drawn on every 2 subintervals.

The function is being partitioned by {a= x0, x1, x2,…,xn=b} in to 2N sub-intervals (must be even) of equal length

2.19. error of composite Simpson’s Rule

Or

|| | |

The proof is on Thomas Calculus P.610

5. Fixed-point iteration

It is used to solve equations of the form E: x-g(x)=0

2.20. theorem:

Suppose g(x) ,not x-g(x) satisfies:

(1)g(x) is continuous and differentiable on the interval [a,b]

(2) , - , - (3) ,-|| Then for any choice of , -, the sequence*+, generated by

for , converges to a point , -

Convergent cases for ___

Convergent cases for ___

special case : for g(x)=1/kx, it forms a loop –

Graph miss chan 4-3

graph miss

chan 4-2




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2.21. method

1. Rearrange the equation E: f(x)=0 in the form x=g(x), where g(x) is the iteration function

2. Use the initial value x0, obtain a sequence of x1, x2,…, xn such that xn is very close to r by substituting them

in the formula

3. Then a fixed point r is obtained such that

IMPORTANT NOTE

In changing f(x)=0 to x-g(x)=0(NOT f(x) to x-g(x)), choose g(x) smartly

example: f(x)=2x-1

g(x)=3x-1 fails

g(x)=(2x+1)/4 works

2.22. error of fixed-point iteration

| |

| |

where is the error of the nth approximation Xn

NOTE

The formula for error may not need to be memorized. There are several formulae. Candidates are required to

derive the formula by using either Taylor’s Expansion with degree zero or Mean

Value Theorem. Please notice that Taylor’s Expansion has a similar structure with Mean Value Theorem when

the degree is zero.

e.g. start by | | || and use mean value theoremNOTE When using calculator, x0 is initially inserted into Ans. After setting up a function in Ans, press [Ans]

one time(doing x1=g(x0)), the value for x1 arises. i.e. press n times, xn appears.

NOTE When “ r is the root of E 1:x-g(x)=0 ” is stated in the question, candidates should bear in mind that

r=g(r).

NOTE To show E1 : x-g(x)=0 has exactly one root in , -, show that

i. f(a)f(b)<0 (positive at one end and negative at the other end)

ii. f(x) increasing/decreasing (not necessarily to have “strictly” increasing/decreasing. A normal function do

not satisfy f(x)=0 for a continuous range of x.

If f(x) has an extreme point within the interval, check it more carefully and adjust the argument to fit the

requirement.




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6. Newton’s Method

2.23. theorem

To solve f(x)=0, use the formula

where the initial point x0 is given

The Newton’s Method is a special case of the fixed-point iteration with the iteration function 2.24. error of Newton’s Method

| | | |

where

1.

Consider Taylor’s expansion about x=r up to degree one with error consideration. ,-,-

7. Secant method

2.25. theorem


where the slope of the secant replaced the slope of the tangent f ’(x) in Newton’s Method

NOTE Please notice the relationship of the numerator and the denominator.

NOTE The formula cannot be used with the replay function in calculators. Candidates have to do it on their own

or by means of calculator program.

NOTE Secant method has not appeared in HKALE since 1993(2009) as secant method is absorbed into the

method of false position and the manipulation in secant method is heavy. Also, the method of false position has

a higher convergent rate. Therefore, secant method is not frequently asked alone.

Kim’s method (should not be used in exam papers)

Use a pair of which are close in secant method to approximate the tangent line (simulation of Newton’s

method). The convergent rate of this simulation should be faster than that of the ordinary secant method.




8. The method of false position (Regula Falsi Method)

2.26. theorem


NOTE Please notice the relationship of the numerator and the denominator.

NOTE The formula cannot be used with the replay function in calculators. Candidates have to do it on their own

or by means of calculator program.

NOTE The method of false position combines the features from the bisection method and the secant method.

The bisection method is effective but not efficient as it converges slowly.

2.27. method

1. Show that a root r of the equation y=f(x) lies in the interval [x0,x1]

2. Compute xn according to the formula where n=1,2,…

3. If f(xn+1) and f(xn) are of different signs, put xn-1=xn+1, otherwise xn=xn+1

NOTE The above statement may be difficult to be absorbed. It could be understood as “ when f(x n+1 ) is positive,

replace x n or x n+1 whose function value is positive and vice versa ”.

4. If | xn+1-xn|<error tolerance, then r= xn+1, otherwise increase n by 1 and repeat steps(2) and (3) until |

xn+1-xn|<error tolerance.

NOTE ---------------------(bracket not converges to zero??)

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