13 Flashcards
A collection of random variables are independent if?
We pick any subset of RV and if we ask what is the probability that the first one is less and equal to some number and/or another one is less and equal to some other number, then its essentially a product of the individual probabilities.
Collection of random variables are identically distributed if?
If the cumulative distribution of one random variable is equal to the cumulative distribution of the other random variable. This means that they all have the exact same probability distribution.
Collection of random variables is i.i.d if?
They are both independent and identically distributed.
Samples taken from a finite population without replacement are not?
i.i.d random variables.
If the sample size is small, computations can be done by?
Treating them as if they were i.i.d random variables. The difference in computed values will be small.
Samples taken from a finite population with replacement are?
i.i.d random variables
The procedures in the following sections assume that the sample taken comes from?
i.i.d random variables.
!!!Steps of sampling to make estimate of population?
Frequency of an interval?
The number of times an outcome (dataset) falls inside the interval (a, b] in n trials. PROBABILITY
Frequency distribution of a dataset?
A table that contains two lists: classes of numerical data and the frequency of each class. The classes are disjoint intervals that cover the entire range of a dataset.
Frequency histogram and axis? 2•
A graph consisting of bars drawn adjacent to each other.
•Horizontal scale represents classes(bins) of numerical data values
•vertical scale represents frequencies.
!!!Classes (bins)?
Numerical data values, independent variable.
Relative frequency of an interval (a, b]?
Proportion of times an outcome falls into the interval (a, b] in n trials.
Sample density of an interval (a, b]?
Relative frequency of the interval is divided by the length of the interval (a, b]. A sample density is computed so that the area of each bar represents the proportion of data within the corresponding interval.
Density histogram? 4•
•x axis: classes(bins)
•y axis: densities
•Heights of bars: sample density values
•area of bar: probability
If you don’t know the probability density function, how do you find the probability of an event?
You have to estimate the probability of the event
If we get subsections of a probability distribution into several rectangles that estimate the probability from a probability density function and decrease the width of the rectangles as it gets smaller and smaller, then it looks like?
A histogram of each rectangle.
As you increase the sample size and plot the histogram of the estimates of a distribution, then?
The histogram starts looking more like the distribution
What happens if you make the bins in a histogram too small?
You hide the distribution as it looks more uniform.
What happens if you make the bins too big?
Then you will hide the distribution
Will a Q-Q plot give you a picture that looks like a density curve?
No
Q-Q plot stands for?
Quantile-Quantile plot
We don’t know the actual distribution but what still exists?
The guess for quantile of population, Q_p, 0<p<1, and the estimate of quantile (Q^_p), estimator for Q_p
A good quantile guess is when?
Q^_p is close to the actual value, Q_p
Quantile-quantile plot?
Graph that plots the estimated quantiles of a population (based on a given sample size n) against the corresponding theoretical quantiles of a given population or the estimated quantiles of a second population (based on a given sample size of m).
One big assumption of I.I.D?
Each sample must be independent, so you must have replacement in any sample size or you can have no replacement in small sample size.
