Mathematics: Statistics 2 (Notes 1 of 1)

Continuous random variables can take any value in a given range. A graph showing how likely X is to take a given value can be drawn and is called a probability density function (pdf). The area under a pdf shows the probability - for this reason the probability that the variable takes a certain range of values is calculated because to work out the probability the variable equals a particular number you would have to do the area of a strip with no width which would be meaningless.The total area under a probability density function is 1 as it is just the total probability of the random variables taking one of its possible values. The probability can never be negative so the graph will always be above the x-axis. To find values for example the upper value or the lower value that the function can take, integrate and set the probability equal to 1. To find the value of the median integrate between the lower bound and m or m and the upper bound and set the function equal to 1/2.Find probabilities by finding the area under the curve, between the lower bound and that point - either using geometry or by integrating.The mean (\(\mu\)) or expected value is given by \(\int xf(x) dx\) between plus and minus infinity. The variance is given by \(\int x^2 f(x) dx\) - \(\mu^2\).  

The normal distribution is bell-shaped. This means things are most likely to be 'somewhere near the middle' and far less likely to just take extreme values. The normal distribution is a special type of continuous distribution (and the total area under a pdf is 1).The larger the variance the shallower the graph, the smaller the variance the taller and steeper the graph will be.If X is normally distributed with mean \(\mu\) and standard deviation \(\sigma\) it is written as X~N(\(\mu\),\(\sigma^2\)).The standard normal distribution has mean 0 and variance 1, ie. X~N(0,1).To calculate probabilities the value will often have to be 'standardised', this means you find the value on the standard normal distribution (Z) which corresponds to the position of the given value on it's normal distribution. To do this subtract \(\mu\) and divide by \(\sigma\). You can then look up the Z value in the normal tables in the formula book. The probability given is the probability of a randomly chosen member of the population being less than the Z value. So to work out the probability that a number is bigger than the Z value subtract the value in the table from 1.It is always useful to draw a sketch of the normal distribution to remind you exactly which probability you are trying to find. Because the normal distribution is symmetrical then the probability of something being less than the mean or bigger than the mean is 0.5, you can use this fact to check that the answer you get makes sense. For example, if the mean is 50, the probability that a value is less than 48 MUST be less than 0.5.You can also use the tables to find the z value if you're given a probability, by looking in the main body of the table first. Alternatively, if the probability is a 'critical value' for example 0.99, 0.95, 0.999 then there's the critical value table which will give the z value. If the population parameters are given you can then use 'standardising in reverse' which means you multiply by \(\sigma\) and add \(\mu\).  You can also find probabilities or z values on certain calculators. The Normal Distribution can also be used to model the binomial distribution under certain conditions:If X ~ B(n,p) where np>5 and nq>5 then the approximation is X ~ N(np,npq)Because of the shape of the binomial distribution this works best when n is large (n>30) and p is approximately 0.5 (so the distribution is approximately symmetrical). A continuity correction also needs to be applied, because the binomial distribution is discrete and the normal distribution is continuous. This means you add or subtract 0.5 from the value to include or exclude the given value (depending on whether it is a greater than and equal to or less than and equal to, or a greater than or less than question). The normal distribution can also be used as an approximation for the Poisson distribution (on the next page).

The Poisson Distribution models the number of events which happen in a particular time period. Something will follow this distribution if:-The events occur Randomly -The events occur Independently - one event doesn't influence the probability of a subsequent event happening-The events occur Singly - one at a time-The events occur at a Constant rate - so the expected number of events that occur  is proportional to the length of the period The distribution has just one feature, \(\lambda\), which is the amount of times an event happens in a given time interval - it is a discrete distribution because either the event happens or it doesn't, ie. it can happen once or twice but not 1.2 times. The formula isThe feature (parameter) \(\lambda\) is the mean and variance, so the standard deviation is the square root of \(\lambda\). Poisson distributions can be combined or scaled up if appropriate:If X~Po(\(\lambda\)) then the number of events in \(x\) lots of time ~Po(\(x\)\(\lambda\))If X and Y are independent variables X~Po(\(\lambda\)) and Y~Po(\(\mu\)) then X+Y~Po(\(\lambda\)+\(\mu\))There are cumulative Poisson tables which work in exactly the same way as cumulative binomial tables for various small values of \(\lambda\). Poisson tables can also be used in reverse to find the smallest value of \(\lambda\) or the smallest value of n which will lead to a particular probability. Poisson distribution can be used as an approximation for the binomial distribution x~(n,p) if n is large and p is small (n>50 and npNo continuity correction is required because both binomial and poisson are discrete distributions. If \(\lambda\) is large (\(\lambda\)>15) then tables and the equation aren't appropriate so an approximation might be appropriate, so a normal approximation can be used. giving X~(\(\lambda\),\(\lambda\)).A continuity correction (adding or subtracting 0.5 as appropriate) is required because Poisson is a discrete distribution and Normal is continuous.

A population is a group of people or items which is being investigated; it may be impractical or expensive to collect information from every single member of a population so information is collected from a selected few members of the population, this is a sample. If a sample is going to be used it is important that it is representative of the population as it is going to be used to draw conclusions about the whole population. To ensure it is not biased a random sample is taken - a random sample is one in which each possible sample of that size as an equal chance of being chosen. To take a random sample:-Obtain a list of each item in the population.-Allocate a number to each item on the list.-Use random number tables to select items from the list until the chosen sample size ha been taken.If a population follows a given distribution that doesn't mean that any one sample will follow the same distribution. However, if all the possible samples of a given size are taken and their means calculated then the distribution of the sample-means (called the sampling distribution) will follow the same distribution as the population. Even if you don't know anything about the distribution of the population the Central Limit Theorem can be used to approximate a normal distribution for a sample provided that n (the number of items in the sample) is large - assume n>30 is enough. If a sample of n readings is taken from a population from mean \(\mu\) and variance \(\sigma^2\), then the sampling distribution is distributed with mean \(\mu\) and variance \(\frac{\sigma^2}{n}\).

Continuous Random Variables

Normal Distribution

Poisson Distribution

Sampling and Hypothesis Tests