Econometrics Exam #1
For join probability distribtuon what must all hte combinations of probabilites sum to? How do you write the joint probability distribtuion?
msut sum to 1 written as the function P(X=x, Y=y)
can yo uuse pmf to give you the probabiltiy distribtuion for a continous r.v.?
no, pmf's are only used for discrete r.v.'s
How do cumulative probabiltiy distribution and probability density functions show the same infromation in different formats?
probability density function: the probability that the commute takes between 15 and 20 minutes is givwn by the area under the pdf between 15 adn 20 minutes which is 0.58, or 58% cumulative probability distribution: the difference between the probability that the commute is less than 20 minutes (78%) and the probability that it is less than 15 minuts (20%) is 58% the same answer that the pdf determined
What do random variables for with information?
r.v. summarize the information
What does this symbol Ω stand for?
represents sample space
The set of all possible outcomes is called the
sample space
What is the F distribution?
the F distribution with m and n degrees of freedom, dneoted Fsubm,n is defiend to be the dsitribution of the ratio of a chi-sqaured rando mvariable with degrees of freedom m, divided by m, t an indepdnetly distributed chi-squared random variable with degrees of freedomn, divided by n
What is the chi-squared distribution?
the chi-squared distirbution is the disitrbution of the sum of m squared independent standard normal random variables
What is an event?
An event is a set of outcomes an event is a subspace of the sampel space eg- If you flip a coin once the sample space is Ω = {H, T} and some events could be: A = {H} B={T} C={H, T}
Which type of functions odo pmf's give the probability distribution of?
pmf's give the probabiltiy distribtuion of a discrete random variable
what is the probability of an outcome?
the proportion of the time that the outcome occurs in the loing run. if the probability of your computer not crashing while yo uare writing a tem paper is 80%, then over the ocurse of wiritng many term papers you will complete 80% without a crash
What is the rth moment of a random variable?
the rth moment is the expected value of Y^r ex: Skewness = E[(Y-µsuby)³]/σsuby³ is a function of the first, second, and third moments of Y
What is a sample space?
the sample space is the set of all possible outcomes so if yo uflipped a coi noen time the sampel spac eowuld be: Ω = {H, T}
Probability: what is the sample space?
the sample space is the set of outcomes of the experiment ex: coin flip sample space = {H, T} a set is just a list
The standard deviation of a random variable Y, denoted σ subY, is ?
the square root of the variance
What is conergence in probability (aka consistency)
the property that Y bar is near µsubY with increasing probability as n increases the law of large numbers states that, under certain conditions, Y bar converges in probability to µsubY or equivalently that Y bar is consitsten for µsubY
What is the student t distribution?
the student t distirbution with m degrees of freedom id defined to be the distirbution of the ratio of a standard norma lrando mvariable, divided by the square root of an indpdenelty disdtrubed chi-sqaured rando mvariable with m degrees of freedom divided by m Let Z be a standard nroma lrandom varibale, let W be a rando mvariable with a chi-sqaured distribtuion with m degrees of freedom, and let Z and W be indepdnendly distirubted the rnadom variable Z/√(W/m) has a student t distribution with m degrees of freedom denoted tsubm
If the conditional mean of Y des not depend on X, then ?
then Y and X are uncorrelated if E(Y|X)=µsuby, then cov(Y,X)=0 and corr(Y,X)=0
What does n∑i=1 (xi-x bar) = ? and what does it mean?
(x₁-x bar)+(x₂-x bar)+....+(xₙ-x bar) = n∑i=1 xi - n∑i=1 x bar = (x₁+x₂+...+xₙ) - x bar n∑i=1 1 = n∑i=1 xi - (x bar × n) = nxbar-nxbar =0 the sum of deviations from the average is zero ex: x₁=1 x₂=6 x₃=5 x bar=4 (1-4)+(6-4)+(5-4) = -3+2+1 = 0
All random variables satisfy the following:
1) they take numeric values (r.v. has to turn outcome to a single number) 2) All r.v.s have a probability distribtuion
What are the four important propertiesof the multivariate normal dsitirbuton?
1. if X and Y have a bivaraite normal distirbution with covariance σsubXY and if a and b are two constantd, then aX+bY has the nroma ldistirbution: N(aµsubX+bµsuby, a²σ²subX+bσ²subY+2abσsbXY) if n random variables have a multivariate normal distirbution, then any lienr combination of these variables (such as their sum) is normally distirbuted 2. if a set of variabels has a multivariate norma ldsitiebutiion, then the marginal distribution of each of the variables is normal 3. if variables with a multivarite nroma ldisitrubtion have covariances that equal zero, then the variables are indepdnendent 4. If X and Y have a bivaratite nromal distribution, then the condtional expectation of Y given X i linear in X; that is E(Y|X=x)=a+bx
What are the properties of pmf?
1. pmf has to be between 0 & 1 0≤g(wₙ)≤1 for n=1, k 2. The probabilites have t osum up to 1: from n=1 to k ∑ g(wₙ) = 1
Two key tools used to apprixamte sampling sitrbtuions when the sample size is large are:
1. the law of large numbers 2. the central limit theorem
What is a porbability distribtuion?
A probability distribution of a random variable Q is a function that has all of the information we need to answer questions about the probability of any event of Q
If Q is the exact number of seconds for the quarter to stop spinning what are some exampels of events of Q?
A= [5.3, 7.5] B=(1.5, 2) C={8.210000....} etc. C is a set that contains one number These events are subset of the sample space
Let X be a discrete random variable with probabiliy mass fucnton f(x). X takes values in the set {w₁, w₂, ...., wsubk} Then, the expected value of X is ?
E(X) = ∑ from j=1 to k w subj f(w subj) = ∑ from j=1 to k w subj P[X=w subj]
What is the law of iterated expectations?
E(Y)=E[E(Y|X)] the inner expectation o nthe right-hand side of the equation is computed using the conditional distributio of Y given X and the outer expectation is computed using the marginal sitribtuion of X ex: the mean number of computer crashes M is the weighted average of the conditional expectation of M iven that th ecomputer is old and the conditona lepxectation of M given that the comuter i new E(M)=E(M|A=0)×P(A=0)+E(M|A=1)×P(A=1) =0.56×0.50+0.14×0.50 = 0.35
What is the mean of Ybar? the variance of Ybar? standard deviaiton of Ybar?
E(Ybar)=µsubY var(Y bar)=σ²subYbar = σ²subY/n std.dev(Y bar)=σsubYbar=σSubY/√n
What is the formula of kurtosis?
E[(Y-µ suby)⁴]/σ⁴ suby
If a distribution is symmetric what does E[(Y-µsuby)³] equal ad why? If a distribution is skewed left what does E[(Y-µsuby)³] equal ad why? If a distribution is skewed right what does E[(Y-µsuby)³] equal ad why?
E[(Y-µsuby)³]=0 if symmetric because the positive values of (Y-µsuby)³ will be offset on avergae by equally likely negative values E[(Y-µsuby)³]= negative value if the distribution has a long left tail because the negative values of (Y-µsuby)³ are not fully offset by the positive values E[(Y-µsuby)³]= positive value if the distribution has a long right tail because the positive values of (Y-µsuby)³ are not fully offset by the negative values
What is the cumulative distribution function of a random variable Q?
F subQ(x)=P(Q≤x) for all real numbers x
How can you have a discrete r.v. even though the sample space is continuous?
If you define your r.v. as T(ω)=1 if ω>5 and 0 otherwise then you made your r.v. discrete
What is the formula for the f distirbution?
Let W be a chi-squared rando mvariable wit hm degrees of freedom and let V be a chi-squared random variable with n degree of freedom, where W and V are independelty distributed (W/m)/(V/n) has an Fsubm,n distirubtion- that is an F distirubtion with numerator degrees of freedom m and denomiantor degrees of freedom n
How is the normal distribution denoted?
N(µ, σ²)
For continous random variables what is P(Q=q₁) equal to?
P(Q=q₁) = 0
The conditonal probabiliy that Y takeso nthe vlaue y whe nX takes o nthe value x is written:
P(Y=y | X=x)
What is the formula for independence?
P(Y=y | X=x) = P(Y=y) X and Y are independent is the conditonal distribution of Y given X equals the margina ldistribuion of Y. or P(X=x, Y=y) = P(X=x)P(Y=y) the joint distribution of two indpdentndent random varibales if the product of their marginal distributions
Formula for conditional distrbution of Y given X=x is
P(Y=y|X=x) = P(x=x, Y=y)/P(X=x)
What is a sampling distribution?
Since Y bar is random, t has a probability distirbution. The ditribution of Y bar is called the smapling distribution of Y bar becasue it is the probabiltiy distribution asssocied eith posbbile vlues of Y bar that could be computed for different possible sampels Y₁.....Yₙ
What is the formula for skewness?
Skewness = E[(Y-µsuby)³]/σsuby³ where σsuby is the standard deviation of Y
What is a probability mass function?
Suppose we have a discrete rando mvaribale W than can take k different values of w We define the probability mass fucntion (pmf) as a function g such that g(wₙ)=P(W=wₙ) = n=1, k g(w₁)=P(W=w₁) g(w₂)=P(W=w₂) ..... g(wk)=P(W=wk)
What is an exact istirbution or finite sample distribution?
The sampling distirbution thatexactly descirbes the distibution of Y bar for any n The exact approach entials derivign a forumla for the sampling distirbution that exactly holds for any value of n
What does it mean for a distribution to be leptokurtic?
a dsitribution is leptokurtic (aka heavy-tailed) if its kurtosis exceeds 3 because a normal distribution has a kurtosis of 3
What is independence?
Two random variables X and Y are independent if knowing the value of one of the variables provides no information about the other X and Y are independent is the conditonal distribution of Y given X equals the margina ldistribuion of Y or the joint distribution of two indpdentndent random varibales if the product of their marginal distributions
What is the sample average (sample mean)?
Y bar of the n observations Y₁...Yₙ is Y bar = (1/n)(Y₁+Y₂+....+Yn)=(1/n)∑ from 1 to n Ysubi the value of Y bar differs fro one rnadomly draw nsmaple to the next Y bar is a rando mvaribale becasue the value of eac hYsubi that comprie are randomly selected
Can you use the probability distribtuion to compute the probability of an event?
Yes. the probability of the event of one or two computer crashes is the sum of the probabiltiies of the consitutent outcomes. That is: P(M=1 or M=2) = P(M=1) + P(M=2) = 0.10+0.06=0.16 or 16%
For random sampling, what does it mean to be identically distirbuted?
Y₁....Yₙ are said to e identiclaly dsitibuted whe nYsubi has the same margina ldistirbution for i=1,....n this is becasue Y₁....Yₙ are radnomly drawn frorm the same population, the marginal dsitirubiton of Ysubi is the same for each i=1....n this margina ldistirbution is the distirbution of Y i nthe population beign sampled
What is a cumulative probabiltiy distribution (cumulative distribtuon function, cdf)?
a cdf is the probability that the random variable is less than or equal to a particualr value the probability of at most one computer crash, P(M≤1) is 90% which is the sum of the probabilities of no computer crashes, 80%, and one computer crash, 10%
What is a nrmal distributon?
a continuous random variable wit ha nromal distirbution has the fmailar bellshaped probabiltiy density the normal density with mean µ and variance σ² is symmetric around its mean and has 95% of its probability between µ-1.96σ and µ+1.96σ skewness is zero and kurtosis is 3
What is a discrete random variable?
a discrete random variable is a random varibale that takes a finite or countably infinte number of values eg- flipping a coin until we get a heads we would be able to count until we recicved a head
What is the differnce between a discrete random variable and a continuous random variable?
a discrete random variable takes on only a discrete set of values, like 0, 1, 2 whereas a containuous random variable takes on a continuum of possible values
What is a probability density function?
a function used for continuous random variables, where the area under the pdf between any two points is the probabiltiy that the random variable falls between those two points example: the probability that the commute takes between 15 and 20 minutes is givwn by the area under the pdf between 15 adn 20 minutes which is 0.58, or 58%
What is the kurtosis of a distribution?
a measure of how much mass is in its tails, and herefroe, is a measure of how much of the variance of Y arises fro mextreme values
What is a probability density function (pdf) ?
a probability density function of a random varibale is a function such that P(a<X<b) = ∫ from a to b f subx(z)dz
When is a random variable a continous random variable?
a r.v. is a continuous r.v. if P(Z=a)=0 for all numbers a
A numerica lsummary of a random outcome is what?
a random variable the number of times your computer crashes while you are writing a term paper is radnom and takes on a numerica lvlaue, so it is a random variable
Probability: What is a random variable?
a random variable is a function that translates each outcome to a number outcome→random variable→numbers ex: Define the random variable X to be 1 when the outcome is H and 0 when the outcome is T example of function: f(x)=x² f(2)=4 X(H)=1 X(T)=0 input is either H or T output is either 1 or 0 depending on the input you define what the random variable equals you could say X(E)=-72.3 if you wanted
What is simpe random sampling?
a sampling scheme in which n objects are slected at random from a popualtion (the populatio of commuted days) and each member of the popualtion (each day) is equally likely to be included in the sample
An event of a random varibale x is ?
a set of values that x can take
What is a Bernoulli random variable?
a special case of discrete rnadom variables whe nthe random variable is binary- the outcomes are 0 or 1 the bernoulli distribtuion is the probability distribtuion of a bernoulli random variable example: let G be the gender of the next new person you meet ,where G=0 indicates that the person is male and G=1 indicates that the person is female the outcomes of G and their probabilites thus are: G= {1 with probaility, p {0 with probability, 1-p where p is hte probabiltiy of the next new person you met being a woman
What is an outlier?
an extreme value of Y
What is conditional distribution?
conditional distribution of Y given X is the distriubion of a random variable Y conditonal on another random variable X taking on a specifc value Ex: What is the probability of a long comut if yo uknw oit is raining? The join probability of a rainy short commute is 15% and the joint probability of a rainy long commute is 15%, so if its raining a long commute and a short commute are eqaully likely. Thus the robability of a long commute conditional on it beign rainy is 50%
What is the formula for correlation?
corr(X,Y) = cov(X,Y)/√(var(X)var(Y))
How is covariance denoted? What is the formula for covariance?
dnoted cov(X, Y) or σsubXY σsubXY = E[(X-µsubX)(Y-µsuby)] = sum from i to k sum from j to l (xsubj-µsubX)ysubi-µsubY)P(X=xsubj, Y=ysubi)
If your events ar defiend as: A = {H} B={T} C={H, T} then which events happened if you flip a T?
event B and event C happened C does not represent flipping the coin twice; C corresponds to one event (flipping a coin) C therefore will always occur when you flip a coin
summation operator: p∑i=1 f(xi) = ? 3∑i=1 i²=? 3∑i=1 a=? n∑i=1 a=?
f(x₁)+f(x₂)+.....+f(xp) 1²+2²+3² = 14 a+a+a = 3a na
What is skewness of a distribution?
how much the distribution deviates from symmetry
What is the central limit theorem?
hte central limit hteorem says that, under general condiotns, the distribution of Y bar is well approxiamted by a normal dsitirbution when n is large Accoridng to the central limit theorem, when n is large, the distirbtuion of Y bar is aproximately N(µsubY, σ²subYbar) the dsitribtuion of Y bar is exactly N(µsubY, σ²subYbar) when the sample is drawn from a population with the normal distirbtuion N(µsubY, σ²subY) The central limit theorem says that this same result is approximetly true when n is large even if Y₁....Yₙ are not themselves nromally distirbuted
Why is it called a "random" variable
it is called a rnadom variable not becasue the function is random but because the input of the fucntion is random eg- the outcome of a coin flip being H or T, whihc is the input for the function, is randomly decided
What does in mean when Y₁.....Yₙ are independently and idnetically distirubted?
it means that Y₁.....Yₙ are drawn fr mthe same distirbution and are indendldenly distirbuted
in chi-squared sitributions what is m?
m is clled the degree of freedoms of the chi-squared distribution ex: Z1, Z2, and Z3 be indepdnent standard nromal rando mvariables. Thn Z₁²+Z₂²+Z₃² has a chi-squared distribution wit h3 degrees of freedom a chi-sqaured distribtuion of m degrees of freedom is denoted Xsubm²
IF the covariance is postiive do X and Y move in the same direction or in diffrent directions? IF the covariance is negative do X and Y move in the same direction or in diffrent directions? IF the covariance is zero do X and Y move in the same direction or in diffrent directions?
move in same dirction move in different directions X and Y are independent
What is conditional variance?
the conditional variance of Y given X is the variance of the conditonal distribution of Y given X the expected number of computer craahes for new computers (0.14) is less than that for old ocmputers (0.56), and the spread of the distribution of the number of crashes, as measured by the conditonal standard deviaiton, is smaller for new computers (0.47) than for old computers (0.99)
What is the conditiona lexpectation of Y given X (also called the conditional mean of Y given X)
the conditona expectation of Y given X is the mean of the conditional distribution of Y given X
What is correlation?
the correlation between X and Y is the covariance between X and Y divided by their standard deviations the random variables X and Y are said t obe uncorrelated if corr(X,Y) =0 the correlaion is always between -1 and 1
The probabilities with whic ha rnadom variable takes on different values are summarized by ?
the cumualtive distribution fucntion the probabiltiy disstribution fucntion (fro discrete random variables) the probability density fucntion (for continuous random variables)
Probability: what is an outcome?
the end result of an event ex: when i flip a coin, there are two outcomes that can happen. Either the coin turns up heads or tails
For the comouter crash example: M is the number of computer crashes if 0 crashes has a probabiltiy of 80%, 1 crash 10%, 2 crashes 6%, 3 crashes 3%, and 4 crashes 1% then what is the Expected value of M?
the expected value of M is the average number of crashes over many term papers, wieghted by the frequency with which a crash of a given size occurs E(M) = 0(0.80)+1(0.10)+2(0.06)+3(0.03)+4(0.01) = 0.35 the expected number of computer crashes while writing a term paper is 0.35 times. in real life this has to be an integer- y ucant say a computer crashed 0.35 times but E(M) represents the average number of crashes over many such term papers
What is the expexted value of a random variable Y?
the expected value of a random variable Y is the long run average value of the random variable oer many repeated trials or occurences Denoted E(Y) or µ suby
The variance, denoted var(Y), of a random variable Y is ?
the expected value of the square of the deviation of Y fro mtis mean: var(Y) = E[(Y-µ subY)²] = from n=1 to k ∑ (yₙ-µ subY)²p subn
What is Covariance?
the extent to which two random variables move together The covariance between X and Y is the expected value E[(X-µsubx)(Y-µsuby)] where µsubx is the mean of X and µsuby is the mean of Y
What is joint probability distribution?
the join probability distribution os two discrete random variables, say X and Y, is the probability that the random variables simultaneously takeo n certain values, say x and y the joint probability distribution is the frequency with which each of these four outcomes ocur over many repeated commutes Ex: 15% of the days have rain and a long commute, rai nand short commute is 15%, long commute and no rain is 7%, and short commute no rain is 63%
What is the asymptotic distirubtion?
the large-scale approximation to the sampling stribtuion The approxiamte approach uses the approximations to the sampling distirbution that rely on the sample size beign large called asymptotic becasue the approximations become exact i nthe limit that n→∞ these aproximations can be ver yaccurate eve nif the sample size is only n=30 observations. Since sample sizes used in practice typically number in the hundreds or thousands, these asymptotic distributions can be counted on to provide very good approximations to the exact smapling distiubtions
What si the law of large numbers?
the law of large numbers states that Y bar will be near µsubY with very high probability when n is large somestiems called the law of averages when a large numebr of rando mvariables with the same mean are avergaed togerther, the large values balance the smal lvlaues and thei rsample avergae is close to their common mean
What is the probability distribtuin of a discrete random variable ?
the lsit of all possible values of the variable and the probabiltiy that each value will occur. These probabilites sum to 1. Computer example: Let M be the number of tiems your computer crashes whil you are writign a term paper. The probability distribtuion of the random variabke M is the list of probabilites of each possible outcome the probability that M=0 denoted P(M=0) is the probability of no computer crashes, P(M=1) is the probablity of one computer crash The probabiltiy distribution ofr M is: the probability of no crashes is 80%, the probability of one crash is 10%, the probabiltiy of two, three , and four crases is 6%, 3%, and 1% respectively
For rando msampling how are the observations in the sampel denoted?
the n observatins i nthe sample are denoted Y₁,....Yₙ where Y₁ is the first observation, Y₂ is the second obervation, and so forth Y₁ is the commuting tiem on the first of her n randomy lselected days and Ysubi is the commutign time on the ith of her randomly selected days
What is multivariate normal distributon? What is bivariate normal distributon?
the norma ldistribution can be generaized to describe tje joint disctrubiotn of a set random variable bivariate nromal distributiin iswhen only two variabels re beign consdiered
What is the standard normal distribution?
the normal distribution with mean µ=0 and variance σ²=1 and is denoted N(0,1) random variabels with a standard naormal istirbuiton are ofte ndenoted as Z and the stanard nrmal cumualative dsitrubtion function is denoted by Φ
The mean of the sum of tw orandom varibables, X and Y, is ? The variance of the sum of X and Y is ?
the sum of their means E(X+Y)=E(X)+E(Y)=µX+µY the sum of their variances plus two times their covraince var(X+Y)=var(X)+var(Y)+2cov(X,Y)=σ²subX+σ²subY+2σsubXY
What are the mutually exclusive potential rsults of a random process?
they are called outcomes eg- if you have a computer, it might never crash, it might crash once, it might crash twice, and so on Only one of these outcomes will actually occer which is why the outcoems are mutually exclsuive
What do the standard deviation and variance measure?
they measure the dispersion or the "spread" of a probability distribtuion
How do you standardize the variable?
to look up probabilites for a nroma lvariable wit ha general mean and variance, we msut standardize the variable by first susbtractign the mean, then by dividing the result by the standard deviation ex: Y is distrubted around N(1,4)- what is the probability that Y≤2? (Y-1)/√4 = (1/2)(Y-1) Y≤2→(1/2)(Y-1)≤(1/2)(2-1) (1/2)(Y-1)≤1/2 P(Y≤2)=P[(1/2)(Y-1)≤(1/2)(2-1)]=P(Z≤1/2)=Φ(0.5)=0.691
The event "my computer will crash no more than ovce" is the set containg what outcomes?
two outcomes: "no crashes" and "one crash"
if 0 crashes has a probabiltiy of 80%, 1 crash 10%, 2 crashes 6%, 3 crashes 3%, and 4 crashes 1%, and the mean crashes is 0.35 then what is the variance of the numebr of computer crashes M and standard deviation?
var(M)= (0-0.35)²(0.80)+(1-0.35)²(0.10)+(2-0.35)²(0.06)+(3-0.35)²(0.03)+(4-0.35)²(0.01) = 0.6475 standard deviation of M is the square root of the variance so, σ subM = √(0.6475) ≅ 0.80
What can we use to establish the probability distirbution of a discrete random varibale?
we can use a probability mass function or a cumualtive distribution function
When is there more liely to be outliers i na distrubtuon?
when the kurtosis of a dstirubtion is large
$10 is each persons pocket 35 pople i nthe room what is the total amount of money?
x bar = 1/n n∑i=1 xi 10 = 1/35 n∑i=1 xi n∑i=1 xi = $350
What is x bar?
x bar is the mean or "average" of the xi (some list of numbers) and define it as follows: x bar = 1/n n∑i=1 xi
Under simple random sampling, is Y₁ distributed indepdnetly of Y₂?
yes becausse under simple rando msampling knowing the vlaue of Y₁ provides no information about Y₂, so the conditonal distribution of Y₂ given Y₁ is the same as the margina ldsitirubiton of Y₂
Suppose we flip a single coin two times sequentially. What is the sample space?
Ω={HH, HT, TH, TT}