5
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Problem

A farmer wants a better understanding of rainfall on their field. Assuming rain falls randomly and with equal likelihood over the entire field, the farmer thinks they can model the volume of rainfall $R$ on any given patch as normally distributed: $$R \sim \mathcal{N}(\mu ,\, \sigma^2)$$ ... meaning most areas will receive similar rainfall but some patches will get lots and some not very much. The farmer wishes to understand this distribution so that they can know how much additional watering is required to ensure at least 95% of their crops have water above a threshold. Too much water is not an issue as the soil is very free draining.


In order to determine the parameters of this distribution, the farmer sets up $n$ measuring stations evenly distributed over the field, and over one week measures and records the volume of rain at each location. Here is a representative sample of 10 measurements:

Vol/ml
20
23
13
7
13
16
15
12
6
21

The issue is, the farmer only has access to a relatively crude measuring device, which introduces quite a lot of uncertainty in the measurements. They estimate this uncertainty to be bounded within the range $\pm 5 \mathtt{ml}$ but wish to model the probability distribution of this uncertainty as triangular: $$E \sim \operatorname{triangular}(\mu = 0, \pm 5 \mathtt{ml})$$ ... meaning for example that a recorded measurement of 13ml corresponds to a true value in the range 8 - 18 ml but most likely to be somewhere in the middle according to a triangular p.d.f.

Question

Given the measured values and their uncertainties, what are the parameters $\mu$ and $\sigma$ of the normal distribution? Or put another way, for what range of values can the farmer be 95% sure any part of the field will receive a volume of rainfall within that range.


Attempt

For context, my level of knowledge is Undergrad General Engineering. I think this question is related to the field of inference but I have not covered that

Here are some thoughts. I believe first of all that because the measurement error $E$ is triangular and symmetrical about its mean, the expected value of rainfall within the field is just the mean of the recorded values $$\text{Expected Rainfall} = \sum_{i=1}^n x_i /n$$ However the mean might also include the measurement uncertainty, since adding uncertain data to obtain the mean, I know you convolve the probability distributions. Therefore: $$ \mu = \text{Expected Rainfall} + E_1*E_2*\cdots*E_n$$

... where $E_{\mu} = E_1*E_2*\cdots*E_n$ is the convolution of $n$ triangular p.d.fs. By the convolution property of Fourier transforms and the Fourier transform of a triangle function this gives $$\mathcal{F}\left\{E_\mu\right\} = \mathrm{sinc}^{2n}(f)$$


Separately, I know I can work out the standard deviation directly from the data - more accurately I can use student's t-distribution to estimate the parameters of $R(\mu, \sigma)$ using the sample standard deviation for this small dataset. But that ignores the measurement uncertainty.

Intuitively, I expect the impact of the measurement uncertainty to increase the variance of the farmer's calculated gaussian, or in other words they will have a wider range of values to be 95% certain they are correct about the rainfall, since their recorded measurements might be over/underestimating the true value. But I don't know how to put these together!

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    $\begingroup$ Note that the normal distribution probably isn't the best choice here because it has support on all of $\mathbb R$, but there's (presumably) no patch of field that receives negative rainfall. I believe a gamma distribution is more appropriate, though I'm not certain of that. $\endgroup$ Apr 10 at 15:00

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