ExampleSketch the Venn diagram.
(1)
(2)
Divide (1) by (2) to get
Substitute these into the equation
Thenand from (1)
We can draw the Venn diagram.
Example:andare independent.andFind the possible values of and draw a possible Venn diagram.
Label the intersectionthenandSinceandare independent
or
A possible diagram is shown
]]>From the definition of conditional probability,
Again from the definition of conditional probability, we can express the joint probability by conditioning onto give
Substituting (2) into (1) gives Bayes’ theorem:
If there aremutually exclusive possible outcomes forthen we can write
hence
Bayes theorem gives rise to some surprises. Many people diagnosed with disease are falsely diagnosed. Suppose that one in a thousand adults has a disease. When an individual has a disease, a positive result will be returned 99% of the time, while a positive result will be returned for 2 % of individuals who do not have the disease. Let andthenandsoand
Less that one in twenty positive diagnoses are actually true positives. More than 95% of positives results are false positives.
]]>For the data set 2, 4, 4, 6, 7, 8, 12, 13, 17, 17, 18
and
The list is 11 long.which rounds up to 3 so the lower quartile is the third number:
To find the upper quartilecalculatewhich rounds up to 9 so the upper quartile is the ninth number and
To find the median,so the median is the average of the fifth and sixth numbers:
The boxplot is shown below.
The rectangular box represents the ‘middle’ half of the data set.
The lower whisker represents the 25% of the data with smallest values.
The upper whisker represents the 25% of the data with greatest values.
]]>The equation of the regression line of y on x is given by
whereandWe usually find a last using
The correlation coefficient can be found if necessary to found how good a relationship we have.
You may be given summary statisticsetc but I will illustrate an example from scratch.
Example: Find the equation of the regression line of heightabove ground level against temperatureusing the data in the table.
h 
100 
1000 
1500 
2400 
4000 
5500 
9000 
10000 
14000 
t 
24 
10 
8 
2 
4 
8 
20 
22 
28 
It is a good habit to calculate and write down the summary statistics:
Hence
]]>Example:Find
to two decimal places.



0.07 








0.6 


0.2486 


To find the probability in our tables corresponding towe go down to 0.6 then along to where there is a 0.07 in top row, reading 0.2486 from the table. This is not the answer yet though.
We have just found the blue area above. We have to add to it the unshaded area to the left, which is 0.5, obtaining 0.7486, because the question was to findso we need the whole area to the left. This is a consequence of our particular normal tables. Not all normal tables will require you to do this.
]]>What we are in fact doing by saying, 'given thathas happened' is excluding the possibility thatwill not happen. The diagram above becomes restricted to the setmay still happen, and by looking at the setthe fraction of the setin whichmight still happen is the the size of the intersection divided by the size of the setThis gives us the relationship,]]>
means 'may take values greater thanbut less than'
means 'may take values greater than or equal tobut less than'
means 'can take any real value'
Whatever the interval, all probability distributions have in common that they all integrate to one over the set of values that x may take. We can use this in problem solving.
Suppose it is known that a random valiabletakes a Triangular distribution:
We can findby solving the equation
Then
The graph ofis sketched below. The hollow circle atindicates thatis not a possible value for
]]>