Recap Flashcards
(14 cards)
what is the difference between statistics and probability
Statistics reason backwards. Probability reasons forwards.
What is moments
Moments refer to how the probability mass is distributed.
how to find the k’th moment?
E[x^k]
E[(X - c)^k] = Integral {(x-c)^k f(x) dx} from -infinity to infinity
what is a raw moment?
A raw moment is a moment about the origin, c=0
what is a central moment?
moment about the mean of the distribution, where we therefore have c=E[X]
what are the 4 first moments
expecation
variance
skewness
kurtosis
how do we differ between statistics and estimates?
upper case for statistics, because they are random variables. Lower case for their corresponding estimates.
elaborate on sampling distributions
Consider a statistics. If we sample it, and do this many times, we will get some distribution which we refer to the sampling distribution.
elaborate on MLE
We need the distribution of the variables we are working with. then we fidn the joint probablity densitriy. This is the likelihood function. then we log it because it is easier to work with. then we differnetiate partially for each parameter and solve for 0. this then will arrive at estimators for the population parameters.
It is key to udnerstand that MLE will arrive at estimators for the population parameters. So, if we are working with the normal distribution, we will get estimators for the mean and variance.
how do we define a good estimatpr
We want the sampling distribution to be as concentrated around the true population parameter value as possible.
we have 3 properties:
1) Unbiasedness
2) efficiency
3) Consistency
If unbiased, the expected value opf the estimator yield the true population parameter.
how do we define confidence intervals
P(value inside some region) = 1 - alpha
1-alpha is called the confidence level.
what is type 1 error
rejecting the null hypothesis when it was really true
what is type 2 error
not rejecting the null when it is false
