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I have a set of Random Variables which I'm performing operations on that goes something like this:

$ \begin{alignat*}{5} &E\sim(&&A_1 - &&B_1)\cdot &&(C_1 + &&D) \\ &E\sim(&&A_2 - &&B_2)\cdot &&(C_2 + &&D) \\ &\vdots &&\vdots &&\vdots &&\vdots && \vdots \\ &E\sim(&&A_m - &&B_m)\cdot &&(C_m + &&D) \\ \end{alignat*} $

Note the operations are meant on the R.V.s and not on the distributions.

This arises from multiple observations of an experiment to estimate the distribution of $E$, where $D$ is a systematic error common to all experiments. All of these R.V.s (except $E$ of course) are uniformly distributed. The problem I have is combining all of these distributions to get a single distribution of $E$ when $D$ should have the same value (not merely the same distribution) in all equations.

Put another way, if $D$ was subscripted like the other R.V.s, I could create a distribution of $C_i + D_i, i\in \{1,2,\dots m\}$ without any issues. I could then combine the $m$ distributions into a single distribution for $E$ (using numerical methods). This would not be correct though, as it would allow for a possibility (effectively certainty) that $D$ would take different values in each distribution, effectively cancelling each other out to some degree.

How can I combine these rows correctly? Numerically is fine; I'm using R and the distr package as well as doing it by hand + manual coding.

I apoogise if the question is confusing. I'm not very good at mathematical notation or terminology. I'll endeavour to clean it up if you have any questions, and thanks in advance!

EDIT

Making things more concrete, $A_i$ and $B_i$ are inaccurate time measurements of an object from experiment $i$. The time elapsed is therefore distributed as $A_i - B_i$.

The speed of the object in experiment $i$ is measured as $C_i$, again inexactly. The 'random' measurement error is accounted for in the distribution. The speed estimating device is calibrated, but this again has an error. However, this error is systematic and identical from experiment to experiment.

The object in each experiment travels exactly the same distance. This distance is $E$ and the property I'm trying to determine as accurately as possible.

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    (1) Do you mean that $D$ is a constant? (2) If it were not for $D,$ what would be the equation for $E?$ (3) You say the 'value' of $E$ in one sentence and the 'distribution' of $E$ in another; which is it? Is $E$ another constant? Or is $E$ a random variable for which maybe you want to estimate the mean? – 2017-01-10
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    Thanks for the look! Ok, I'll try and answer your comments. This may take a minute. (1) $D$ has a constant value. It is an instrument that is calibrated, and the error is something like $\pm 2units$. The error is unknown (only its distribution is known), but it won't change significantly from experiment to experiment. So if it reads +1unit in experiment 1, it reads +1 unit in experiment $m$. – 2017-01-10
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    (2) I can't write the equation for $E$ exactly. I can write an equation for each row - convolving and so forth. I then use numerical methods to determine $E$ using the folowing steps: the PDF of E at any value $x$ is proportional to the PDF of row 1 at the same point, multiplied with row 2 at the same point, etc. I then normalise so the new PDF integrates to 1. – 2017-01-10
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    (3) I've changed it. Ideally I want the value of $E$ (on the understanding that all of these R.V.s do have one true realised value; it's just unknown exactly what that is). However, that is not possible, so I deal with the distributions. – 2017-01-10
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    (2) Too vague. Also, is only measurement of $C_i$ affected by error $D$? Or $A_i$ and $B_i$ also? If it were not for calibration error $D,$ would $E_i = (A_i - B_i)C_i$ in each row? – 2017-01-10
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    I hope the edit to my question helps. I'll get back to (2). But yes, if not for $D$, you could say $E_i = (A_i-B_i)C_i$ However, note that $E_i$ is not subscripted either; that is, $E_1 = E_2 = E_3 \dots$ – 2017-01-10
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    But each determination $E_i$ a random determination of the distance, and you hope the mean of $E_i$ is a good estimate of the distance?? – 2017-01-10
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    (2) If I ignore $D$ for now, let me define $F_i$ as $(A_i-B_i)C_i$. Let me say that $f_i$ is the PDF of the R.V. $F_i$ and that $e$ is the PDF of the R.V. $E$ (I hope this isn't confusing). Then $e \propto f_1*f_2*\dots*f_m$. I can approximate this numerically by taking a lot of values over the domain of $e$, calculating the $f_i$s and multiplying them. I then create a function which just interpolates between values to make $ke$ where $k$ is a constant of proportionality. I then finally normalise this so the density integrates to 1 and I have $e$. – 2017-01-10
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    Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/51496/discussion-between-timbo-and-bruceet). – 2017-01-10
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    StackExchange asked me to move from comments to discussion. Hope that's OK. – 2017-01-10
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    I'm putting this back in comments as I just realised MathJax doesn't work in discussions! I only produce a single distribtuion for $E$, not multiple $E_i$s. I then use this as an estimate of the distance. Actually, I provide the mean/median/max and range as well as the PDF, as different readers of the report like to interpret these differently. – 2017-01-10

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