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PERMUTATIONS AND COMBINATIONS

The number of permutations of r objects, taken from

a set of n distinct objects without replacement is

given by

n

r

n!P =

(n-r)!

The number of permutation of r objects, taken from a

set of n distinct objects with replacement is given by

nr

The number of permutations of n distinct objects in a

circle is given by

(n-1)!

The number of possible combinations of r objects,

taken from a set of n distinct objects without

replacement is given by

nr n!C =

(n-r)!r!

PROBABILITY

PRO BA BI L I TY

For two events A and B,

P(A B)=P(A)+P(B)-P(A B)

P(A)=P(A B)+P(A B')

MUTUA L EX C L USI V I TY

Mutually exclusive events cannot occur at the same

time. For two mutually exclusive events, E1 and E2,

1 2 1 2P(E E )=P(E )+P(E )

C O N D IT IO N A L P R O B A B IL IT Y

P(A B)P(A|B)=

P(B)

I NDEPENDENC E

Independent events are events the occurrences of

which do not influence the probability of the

occurrence of the other event.

For independent events,

P(A|B)=P(A) or P(B|A)=P(B)

P(A B)=P(A)P(B)

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RANDOM VARIABLES

For any random variable X,

The expectation, , is given by

E(X)= xP(X=x)

and E(aXbY) = aE(X) bE(Y)

The variance is given by22

Var(X)=E[(X-) ]= (x-) P(X=x) and Var(aXbY)=a

2Var(X) + b

2Var(Y)

The standard deviation, , is given by

= Var(X)

CONTINUOUS RANDOM VARIABLES

N O R MA L D IS T R IB UT IO N

For a random variable X modelled by a normaldistribution with mean and standard deviation

X~N(,2)

S T A N D A R D N O R MA L VA R IA B L E

Letting X~N(,2), the standard normal variable Z is

defined asx-Z= ~N(0,1) and

P(Xx) =x-P(Z )

DISCRETE RANDOM VARIABLES

B IN O MIA L D IS T R IB UT IO N

For a random variable X modelled by a binomial

distribution with n trials and probability of success, p

X~B(n,p)

Its probability distribution is given by

P(X=x)=nCxp

x(1-p)

n-x

Its mean and variance are given by

E(X) = np

Var(X) = np(1-p)

P O IS S O N D IS T R IB UT IO N

For a random variable X modelled by a Poisson

distribution with parameter X~Po()

Its probability distribution is given by- xe

P(X=x)=x!

Its mean and variance are given by

E(X) = Var(X) =

Note also that for two Poisson random variables

X~Po(1) and Y~Po(2),

X+Y~Po(1+ 2)

APPROXIMATIONSApproximations marked are to be continuity corrected

B IN O MIA L T O P O IS S O N

For X~B(n,p)

If n is large (n > 50) and p is small (p < 0.1) such that

np < 5, then X~Po(np)

B IN O MIA L T O N O R MA L

For X~B(n,p)

In is large such that np > 5 and n(1-p) > 5, then

X~N(np,np(1-p))

P O IS S O N T O N O R MA L

For X~Po()

If > 10, then X~N(, )

C O N T IN UIT Y C O R R E C T IO N

These are the ranges, for given probability

distribution functions, to consider when

approximating discrete random variables to

continuous random variables.

P(Xa)

P(Xa)

P(Xa)

a-1 a a+1

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SAMPLING

S A MP L E ME A N

The sample mean from a normal population

of sample size n with mean and variance 2

is given by2

X~N(, )n

C E N T R A L L IMIT T H E O R E M

The central limit theorem states that, for a

non-normal population with sample size n,2

X~N(, )

napproximately, if n is large (50).

UN B IA S E D E S T IMA T O R O F S A MP L E ME A N

For any sample size n taken from a population with an

unknown mean , the unbiased estimator of is given by

x (x-a)x= = + an n

where a is a constant

UN B IA S E D E S T IMA T O R O F S A MP L E VA R IA N C E

For any sample size n taken from a population with an

unknown mean 2, the unbiased estimator of

2is given by

( )2

2 2 2x1 1

s = x - = (x-x)n-1 n n-1

( )2

2(x-a)1

= (x-a) -n-1 n

where a is a constant

HYPOTHESIS TESTING

C O NDUC TI NG A H YPO TH ESI S TEST

Step 1: State the null and alternative hypotheses H0 and H1Step 2: State the significance level,

Step 3: Determine the test statistic to use and its distribution

Step 4: Calculate the p-value for the test statistic

Step 5: Indicate whether or not to reject H0 based on the evidence from the sample

H0 is rejected if p-value <

H0 is not rejected if p-value >

TEST STA TI ST I C S

Normal Population Non-normal Population

2

known 2

unknown 2

known 2

unknown

Sample size is large

n50

2X~N(, )

n

2sX~N(, )

n

by the CLT2

X~N(, )

n

by the CLT2

sX~N(, )

n

Test Statistic Z-test Z-test Z-test Z-test

Sample size is small2

X~N(, )n

T~t(n-1)

Test statistic Z-test t-test