7/30/2019 chi square theorems

1/2

Fact Sheet 2Statistics 2008

Department of Mathematics, IIT Madras

If X1, . . . , X n constitute a random sample, i.e., they are independent and identicallydistributed, then the sample mean is X = 1

n

ni=1 Xi, and the sample variance is

S2 = 1n1

ni=1(Xi X)2.

The variance of the sample is V ar = 2X =1n

ni=1(Xi X)2.

From a finite population of size N, If a sample X1, . . . , X n is drawn where Xi is the ithone drawn, the joint probability density of these random variables is given by

f(x1, . . . , xn) = 1N(N 1) (N n + 1)

We say that X1, . . . , X n constitute a random sample from such a population.

When a normal distribution is used as an approximation to a binomial distribution,each nonnegative integer k is represented as the interval [k 0.5, k + 0.5]. That is, thebinomial probability P(X = k) is computed as the corresponding normal probabilityP(k 0.5 X k + 0.5). (Continuity Correction)

The t-distribution with degrees of freedom has the density:

f(t) = ((+ 1)/2) (/2)

1 + t2

(+1)/2for t R.

The F-distribution with degrees of freedom 1 and 2 has the density :

f(x) =

((1 + 2)/2)

(1/2)(2/2)

12

2/2x(1/2)1

1 +

12

x

(1+2)/2

for x > 0

0 otherwise

Theorem 1 IfX1, . . . , X n constitute a random sample drawn from an infinite population with

mean and variance 2

, then X = E(X) = and V ar(X) = 2

X = 2

/n.

Theorem 2 : Law of Large Numbers For any positive constant c, the probability that X

will take on a value between c and + c is at least 1 2

nc2.

Theorem 3 : Central Limit Theorem If X1, . . . , X n constitute a random sample froman infinite population with mean , variance 2, then as n , the limiting distribution ofZ =

X /

n

is the standard normal distribution.

1

7/30/2019 chi square theorems

2/2

Theorem 4 If X1, . . . , X n constitute a random sample from a normal population with mean, variance 2, then X has a normal distribution with mean and variance 2/n.

Theorem 5 If X is the mean of a random sample of size n drawn from a population of size N

with mean and variance 2, then E(X) = and V ar(X) =2

n

N

n

N 1 .Theorem 6 IfX1, . . . , X n are independent random variables each having the standard normaldistribution, then Y =

ni=1 X

2i has the chi-square distribution with = n degrees of freedom.

Theorem 7 If X1, . . . , X n are independent random variables having chi-square distributionswith degrees of freedom 1, . . . , n, respectively, then Y =

ni=1 Xi has the chi-square distribu-

tion with 1 + n degrees of freedom.Theorem 8 If X1 and X2 are independent random variables, where X1 has chi-square distri-bution with degrees of freedom 1, X1 + X2 has chi-square distribution with 1 + 2 (> 1)

degrees of freedom, then X2 has chi-square distribution with degrees of freedom 2.

Theorem 9 If X and s2 are the sample mean and the sample variance of a random sampleof size n drawn from a normal population with mean and variance 2, then X and s2 are

independent and the random variable Y =(n 1)s2

2has a chi-square distribution with degrees

of freedom n 1.Theorem 10 If Y, Z are independent random variables where Y has chi-square distribution

with degrees of freedom , Z has the standard normal distribution, the t =ZY /

has the

t-distribution with degrees of freedom .

Theorem 11 If X and s2 are the sample mean and the sample variance of a random sample

of size n drawn from a normal population with mean and variance 2, then t =X s/

n

has

the t-distribution with degrees of freedom n 1.Theorem 12 If U, V are independent random variables having chi-square distributions with

degrees of freedom 1, 2, respectively, then F =U/1V /2

has the F-distribution with degrees of

freedom 1 and 2.

Theorem 13If s

2

1 and s

2

2 are the sample variances of a random sample of size n1 and n2,respectively, drawn from normal populations with variances 21 and 22, respectively, then

F =s21/

21

s22/22

=22s

21

21s22

has the F-distribution with degrees of freedom n1 1 and n2 1.

2