ECON 221- 001 Liton Chakraborty Statistics for Economists February 16, 2015

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Formula Sheet for Midterm Exam Winter 2015

I. BASIC STATISTICS x , the sample mean, is an estimate of x

ixnx 1 2xs (or just

2s ), the sample variance, is an estimate of 2x :

1

22

nxx

s ix

The sample standard deviation, sx is the square root of the sample variance:

11

222

n

xnxn

xxs iix

The Standard Error of the Mean or S.E.M. is an estimate of the standard deviation of x :

Standard Error of the Mean = nsMES /... Sample coefficient of variation (CV):

Empirical Rule:

contains about 68% of the values in the population or the sample contains about 95% of the values in the population or the sample contains about 99.7% of the values in the population or the sample

The sample covariance:

Sample correlation coefficient:

II. BASIC PROBABILITY

Probability of A or B: If A and B are Mutually Exclusive:

)()()()()()()( BPAPBorAPBandAPBPAPBorAP Conditional probability of A given B: If A and B are Independent:

)()(

)()()( APBAP

BPBandAPBAP

Joint Probability of A and B: If A and B are Independent:

)()()()()()( BPAPBandAPBPBAPBandAP

100%xsCV

1 2 3

1n

)y)(yx(xsy),(xCov

n

1iii

xy

YX ssy),(xCovr

ECON 221- 001 Liton Chakraborty Statistics for Economists February 16, 2015

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Bayes' Theorem

)()(...)()()()()()(

)(2211 kk

iii BPBAPBPBAPBPBAP

BPBAPABP

The odds in favor of A are:

III. PROBABILITY DISTRIBUTIONS Mean and variance of a discrete random variable:

222

1

2

1{E(x)}-) E(x)()]([ = Var(x))()(

i

N

iii

N

ii XPXEXXPXXE

Jointly distributed random variables X and Y The conditional mean is The conditional variance is The covariance between X and Y is

A. The Poisson distribution

The probability of seeing n events is: !

)Pr(nen

n The variance of a Poisson is equal to the mean, so the standard deviation is the square root of the mean, . B. The Binomial distribution

The probability of a given x is: xnx ppxnxnx )1(!!!)Pr(

The mean of a binomial is np and the variance is np(1-p). If n is large and p is small, the binomial is approximated by a Poisson with np . C. The normal distribution

The pdf for the normal distribution looks like this: 22 2/)(

21)Pr(

xex Where and 2 are the mean and variance as usual. A Poisson distribution can be approximated by a normal distribution of the same mean and variance if is large. A binomial can be approximated by a normal distribution if np and n(1-p) are both large.

The standard normal variate, z, : -X =z

)AP(P(A)

P(A)-1P(A) odds

x)|P(y x)|(yX]|E[YY

X|Y

Y

2X|Y

2X|Y

2X|Y x)|x]P(y|)[(yX]|)E[(Y

x y

yxYX y))P(x,)(y(x)])(YE[(XY)Cov(X,

x y

yxyx y)xyP(x,E(XY)Y)Cov(X,

ECON 221- 001 Liton Chakraborty Statistics for Economists February 16, 2015

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D. The Uniform distribution

The mean of a uniform distribution is: The variance: Where, a = minimum value of x; b = maximum value of x E. The Exponential Distribution The exponential random variable T (t>0) has a probability density function

for t > 0

The cumulative distribution function (the probability that an arrival time is less than some specified time t) is:

IV. RULES ON EXPECTATIONS:

E[X+] = E[X]+ when both and are constants. E[XY] = E[X]E[Y] if X and Y are statistically independent.

var(X + ) = 2var(X) var(X) = E[ X2 ] if E[X ] = 0

var(X Y) = 2var(X) + 2var(Y) - 2 cov(X,Y)

Good luck!!!

2ba

12a)-(b

22

tef(t)