Week 3 Flashcards
What are some other examples of distributions beside normal?
Chi squared, F and T distributions.
What is a t distribution given by? Is it standardised?
(the mean of x - the population mean)/sqrt(sample variance/number of items in sample). This will give a t distribution with number of items-1 degrees of freedom. The bottom half is in fact a standardisation in disguise.
What happens to the standard deviation if we condense a sample to an average? What does the standard deviation become known as in this case?
By taking multiple samples and finding the expected average of the samples it creates a much more squeezed normal distribution, lowering the standard deviation. This becomes known as the standard error.
What is the denotion of a standard normal variable?
Xi, …, xn ~ N (u, sigma^2) IID. this means that x is a variable distributed normally with mean u and deviation of sigma^2.
What will be the sample mean of a normally distributed variable X be expected to be?
The population mean, the sample mean will be (1/N)* the sum of x values. Our best guess is that the sum of x values will be the population mean, as such we will expect the sample mean to be the population mean.
What will the standard error of the mean of a normally distributed variable X be?
the variance of each x value / the number of values. This becomes (1/N)^2 * variance of each x value. The variance overall of the x values is expected to be variance * N, (1/N)^2 * variance = variance/N. Hence the standard error = standard deviation/sqrt(N)
How is a Chi squared distribution created?
Take a group of samples all from the same population, then take a value from each one and square it, and add all the squared samples together. This will create a chi squared distribution with degrees of freedom equal to the number of distributions.
What does a chi squared distribution look like?
A distribution with a large hump at the start and a tail at the right end, there will be no negative values, if there are not many distributions, with more degrees of freedom it will look less like a hump and more like a ski jump.
What will the mean of a chi squared distribution be?
The expected value will be given by the (mean^2 + standard deviation^2) * the number of distributions.
What is the skew of a graph, what about kurtosis? How are they typically distributed for a normal distribution sample?
The skew of a graph describes the asymmetry of the graph, with the skew being negative if it has a left tail, or positive if it has a right tail. It is given by the expected value of (X - the expected X value/mean)^3 and has a value of 0 for a normal distribution.
Kurtosis describes the peakedness of a distribution, it is given by the expected value of (X - the expected X value/mean)^4 and takes the value of 3 *(sqrt(population mean))^4 in a normal distribution. Kurtosis will also typically lead to fatter tails, as well as the sharper peak.
How do we make an F test?
If we produce multiple chi squared distributions, and then divide (the results of one/number of distributions it’s made up of) by (the results of the other/number of distributions it’s made up of), this will produce an f test
What should an F test result be if the null hypothesis is true?
The null hypothesis for an F statistic is that the sample deviations are the same. This will mean that the F test will be given by sample deviation/sample deviation, as such the awnser should be 1.
What is it called when we estimate the population mean to be the sample mean?
Taking the population moment.
What is a central moment?
A moment of a probability variable about the random variable’s mean
What is a cauchy distribution? What is its expected value? What about variance?
A cauchy distribution is given by a standard normal divided by another standard normal. They look like standard normals, but with very long tails.
It has no expected value as the integral doesn’t converge (too easy for the denominator standard normal value to be close to 0, making the result astronomically large). As it has no mean it also has no variance.
