1:2:1 Maths Tuition

Probability Density Function...

Query: A random variable has a pdf of f(x)=k(1-x)², for x between 0 & 2 and f(x)=0 elsewhere. How do I find the value of 'k'?

Response: The key point here is that the integral of f(x) between the limits of 0 and 2 must be 1, as it is a p.d.f.

Please click on the graphic for the solution.

Parts Per Million?

Query: What weight of a particular substance would be needed to produce a 2 parts per million concentration within 10kg of a compound substance?

Response: The ‘part per million’ (ppm) measure is very popular in North America, but only widely used within the scientific community in the UK.

Anyone who has tried to add santising chemicals to a swimming pool or hot tub will be already used to calculating quantities in ppm.

 

1 ppm equates to 1mg out of every 1kg

So, by proportion, this leads to…

 

2 ppm equating to 20mg out of 10kg.

What's A Causal Link?

Query:  Explain, briefly, with the aid of an example, what is meant by a ‘spurious’ or ‘nonsense’ statistical correlation.

Response: High values of the correlation coefficient can sometimes be found between two variables for no apparent reason. This is a particular danger when vast amounts of multi-variate data are mined and analysed.

Sometimes there is a causal link between each variable producing the correlation and a third variable which triggers the large coefficient.

A simple example can be found in crime statistics of a local authority. If you run an analysis between the variables (1) the number of police arrests for street drunkeness (2) the number of places of worship within one mile of the arrest, you'll probably find a significant correlation, but where's the likely causal link?

Each of these variables is likely to be positively correlated with the population of the area in which they occur. High population areas are often a focus for drunkeness as there are a large number of bars and clubs to service the demand and will also have a higher density of places of worship, which would probably account for the 'spurious' correlation found.

I remember reading a paper by a researcher from Cardiff University in which he analysed a large amount of quantitative schools' results and student personal characteristics data and found a significant statistical correlation between the Mathematics results of individual schools and the altitude (i.e. the number of metres above sea level) of the school buildings.

His tongue-in-cheek conclusion was that, if possible, schools in low altitude areas should adopt the strategy of improving their student scores at Mathematics by ensuring that all lessons took place upstairs!

Awkward Pythagoras Theorem Question...

Query: Triangle PQR is right angled at Q. The length of PR = 23.6 cm, the length of PQ is x cm and the length of QR is 6.2 cm longer than PQ. Write down and solve a quadratic equation to find the length of PQ correct to 1 decimal place.

Response: Please click on the graphic below to view the solution.

How Many Cuppas?

Query: A tea urn is shaped like a cylinder with a circular cross-section of diameter 35cm and height 60cm. It is filled with tea. How many cups of capacity 150 cubic cms. can be served from the urn?

Response: The answer is 384 cups. Please click on the graphic below to view the solution.