CAIE A-Level Further Math AS 4.3.1 Fitted Distributions Questions

Practise fitting suitable probability distributions to data, estimating parameters and checking model assumptions with Further Mathematics AS questions and mark schemes.

Exam points

Fit a suitable probability distribution to data and estimate its parameters.

[Maximum number: 3]

A random sample of 50 values of the continuous random variable X was taken. These values are summarised in the following table.

Table for Question CAIE A-Level Further Math AS 4.3.1 Fitted Distributions Questions question 1 — CAIE A-Level Further Math AS

It is required to test the goodness of fit of the distribution with probability density function f given by

f(x)= \begin{cases}\frac{1}{24}\left(\frac{4}{x^{2}}+x^{2}\right) & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise }\end{cases}

The expected frequencies, correct to 4 decimal places, are given in the following table.

Show that a=4.6007 and find the value of b.

$\int_{1.5}^{2} \frac{1}{24}\left(\frac{4}{x^{2}}+x^{2}\right) \mathrm{d} x=\frac{1}{24}\left[-\frac{4}{x}+\frac{x^{3}}{3}\right]$
Integration with correct powers and correct limits
seen.

$a=50 \times 0.092014=4.6007$
AG

$\frac{53}{576}$ or $\frac{1325}{288}$ or 0.092014 or 4.60069 seen

b=11.4211
Or 11.421(0)

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CAIE A-Level Further Math AS 4.3.1 Fitted Distributions Questions question 1