2 statistics review
vad visar _o_
it is the sample space
what do random variables do?
random variables describes samples by using numbers
how do we calculate the probability function? P(1, 2) of rolling a dice.
the probability of getting 1 x the probability of getting 2
= 1/6 x 1/6 = 1/36
is Y a random variable, why why not?
yes, Y = .... where the rabdom variabkes are on the right side. this means that we trabsform these rabdom variables into a new one, namely Y.
explain the probability density function
det är en kurva där arean under är probabilities för värden på x-axeln. värdena på y-axeln är INTE probabilities utan vi behlver titta på arean.
prob(x is between 25 and 20) = prob(x <= 25) - prob(x <= 20)
explain (for random variables)
1) expectations
2) variance
3) covariance
1) measure of the center of the distribution of a random variable (prediction). for each sample, we take the vakue of x1 evaluated on the sample and multiply kt by the probability for x1 to get E[X1] = prob(x1) • value of x1 in sample
2) measures how accurately X1 is predicted by its expectation, E[X1].
it is always ^2 so always positive
var(X1) = (X1 - E[x1])^2
3) indicates the co-movement between two random variables
small variance:
"most samples have similar x1"
E[x1] is a good prediction
the density funtion shows a very high and smal kurva vid 0 men små vingar.
large variance:
x1 is very different across samples
E[x1] is not a good prediction
density curve är kort på höjden och bred på längden, så vingarna är tjockare och längre
standard avvikelse:
roten ur variance
better measure in explaining
varför tar vi expectation av variansen istället för bara variansen?
E[...] istället för (x1-a)^2
to remove randomness
vad är covariance?
a measure of the direction of the relationship between two randol variables. positiv, negativ
vad kan vi dra för slutsats om Y1 och Y2 correlerar? vad kan man inte säga?
ok: Y1 och Y2 är statistically dependent. not other way around
ej ok: Y1 och Y2 can be statistically dependent but still be uncorrelated.
vad är fördelen med random sampling?
det är representativt sample of the population
berätta om processen hur vi lär oss om populationen
➡️vi har populationen
➡️drar ett sample som är RANDOM
➡️vi observe the sample with numbers and we have a realized sample. vi denote dessa med små letters för att visa att vi nu har fått siffror på det
➡️these many numbers that describe the realized sample are called: data set
➡️we use statistical methods which are rules for processing numbers in a data set, and apply these to data.
➡️then we learn about the propertiers of the population
förklara begreppet estimation
estimation är när man gissar på en particular population feature where the guess uses knformation that is contained in a random sample
förklara estimator, estimate, estimanf
genom att apply a rule to a random sample, we get the estimator (ofteb sample average, Ê. The estimator är combines flera random variables och är därmed sjölv en random variable
once we draw the sample, and valuate the estimator at the realized sample gives the estimate. it is a number that we ingerpret as a guess of the estimand.
estimand is the feature of the population we are interested of
vad är method of moments?
it is a strategy of how to collect and combine data into estimates. om vi can define our estimand in terms of expectations of population random variables, we can apply method of moments. expectations are sometimes called moments. so it can also be called method of expectations.
själva method of moments är att replace all population moments (expectations) by the corresponding sample averages and we put hats ^ on everything. so we can use method of moments to come up with new estimators to apply to data to get estimates