<p>The result is due to properties of normal random variables.</p>
<p>First,</p>
<blockquote>
<p>If <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are independently normal, then <span class="math-container">$Z=X+Y$</span> and <span class="math-container">$X$</span> are jointly normal.</p>
</blockquote>
<p>Second,</p>
<blockquote>
<p>If <span class="math-container">$X$</span> and <span class="math-container">$Z=X+Y$</span> are jointly normal, then
<span class="math-container">\begin{equation}
E(X\mid Z) = \mu_X + \frac{\sigma_{XZ}}{\sigma_Z^2}(Z-\mu_Z) 
\quad\text{and}\quad 
\text{Var}(X\mid Z) = \frac{\sigma_X^2\sigma_Y^2}{\sigma_X^2+\sigma_Y^2}
\end{equation}</span></p>
</blockquote>
<p>Applying these to your problem, where <span class="math-container">$X=C$</span>, <span class="math-container">$Y=\sqrt\beta\eta$</span>, and <span class="math-container">$Z=z$</span> with the following statistics
<span class="math-container">\begin{align}
\mu_X &amp; = c(\theta,a)x+k \\
\mu_Z &amp; = c(\theta,a)x+k \\
\sigma_X^2 &amp; = \alpha \\
\sigma_Y^2 &amp; = \beta \\
\sigma_{XZ} &amp; = \sigma_X^2 = \alpha \\
\sigma_Z^2 &amp; = \alpha +\beta
\end{align}</span>
we thus obtain
<span class="math-container">\begin{align}
E(C\mid z) &amp; = c(\theta,a)x+k+ \frac{\alpha}{\alpha+\beta}[z - (c(\theta,a)x+k)] \\
&amp; = (c(\theta,a)x+k)\frac{\beta}{\alpha+\beta} + z\frac{\alpha}{\alpha+\beta}
\end{align}</span>
and
<span class="math-container">\begin{align}
\text{Var}(C\mid z) &amp; = \frac{\alpha\beta}{\alpha+\beta}
\end{align}</span></p>

CC BY-SA 4.0
2024-08-19T18:12:24.100Z
2024-08-19T20:48:55.563Z
2024-08-19T20:48:55.563Z
Herr K.
<p>I am reading a paper by Baron and Besanko (1987) (<a href="https://www.jstor.org/stable/2555639" rel="nofollow noreferrer">https://www.jstor.org/stable/2555639</a>) and I am having a trouble in understanding a particular portion of their math that deals with conditional expectations. Any help is deeply appreciated!</p>
<p>C is the cost function: <span class="math-container">$C = c(\theta,a)x + k + \sqrt{\alpha}\epsilon$</span> where &quot;epsilon is the realised value of <span class="math-container">$\tilde \epsilon$</span> with mean 0&quot;
<span class="math-container">$z=C + \sqrt{\beta}\eta$</span>  (<span class="math-container">$\beta$</span> is the randomness parameter, &quot;<span class="math-container">$\eta$</span> is the realised value of <span class="math-container">$\tilde \eta$</span> with mean 0&quot;)
<span class="math-container">$\alpha&gt;0 \beta&gt;0$</span>
They write:
For the case where <span class="math-container">$\tilde \epsilon$</span> and <span class="math-container">$\tilde\eta$</span> are standard normal, independent random variables, conditional distribution of of cost conditional on z can be written as:
<span class="math-container">$g(C|z) = N(C|(c(\theta,a)x + k)\frac{\beta}{\alpha+\beta} + z \frac{\alpha}{\alpha+\beta},\frac{\alpha\beta}{\alpha+\beta})$</span>
where N(.|m,v) denotes normal density function with mean m and variance v.</p>
<p>so, if I am not mistaken, they mean that <span class="math-container">$E[C|z] =(c(\theta,a)x + k)\frac{\beta}{\alpha+\beta} + z \frac{\alpha}{\alpha+\beta}$</span> and <span class="math-container">$Var[C|z] =\frac{\alpha\beta}{\alpha+\beta}$</span>.
I don't understand how have they calculated this mean and variance. Any help is deeply appreciated. I am absolutely lost here.</p>
<p>I would be grateful if someone who has read this paper would be willing to discuss it with me.</p>

<probability><regulation><contract-theory>
Need help in understanding Baron & Besanko (1987) paper "Monitoring, Moral Hazard, Asymmetric Information, and Risk Sharing in Procurement Contracting
Pc1
# Need help in understanding Baron & Besanko (1987) paper "Monitoring, Moral Hazard, Asymmetric Information, and Risk Sharing in Procurement Contracting

(Asked by: username_0 on 2024-08-18)

I am reading a paper by Baron and Besanko (1987) (<https://www.jstor.org/stable/2555639>) and I am having a trouble in understanding a particular portion of their math that deals with conditional expectations. Any help is deeply appreciated!

C is the cost function: $C = c(\theta,a)x + k + \sqrt{\alpha}\epsilon$ where "epsilon is the realised value of $\tilde \epsilon$ with mean 0"
$z=C + \sqrt{\beta}\eta$ ($\beta$ is the randomness parameter, "$\eta$ is the realised value of $\tilde \eta$ with mean 0")
$\alpha>0 \beta>0$
They write:
For the case where $\tilde \epsilon$ and $\tilde\eta$ are standard normal, independent random variables, conditional distribution of of cost conditional on z can be written as:
$g(C|z) = N(C|(c(\theta,a)x + k)\frac{\beta}{\alpha+\beta} + z \frac{\alpha}{\alpha+\beta},\frac{\alpha\beta}{\alpha+\beta})$
where N(.|m,v) denotes normal density function with mean m and variance v.

so, if I am not mistaken, they mean that $E[C|z] =(c(\theta,a)x + k)\frac{\beta}{\alpha+\beta} + z \frac{\alpha}{\alpha+\beta}$ and $Var[C|z] =\frac{\alpha\beta}{\alpha+\beta}$.
I don't understand how have they calculated this mean and variance. Any help is deeply appreciated. I am absolutely lost here.

I would be grateful if someone who has read this paper would be willing to discuss it with me.

**Tags:** probability, regulation, contract-theory

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## 1 answers

### Answer by username_1 on 2024-08-19. Score: 4 | Accepted answer

The result is due to properties of normal random variables.

First,

> If $X$ and $Y$ are independently normal, then $Z=X+Y$ and $X$ are jointly normal.

Second,

> If $X$ and $Z=X+Y$ are jointly normal, then
> \begin{equation}
> E(X\mid Z) = \mu\_X + \frac{\sigma\_{XZ}}{\sigma\_Z^2}(Z-\mu\_Z)
> \quad\text{and}\quad
> \text{Var}(X\mid Z) = \frac{\sigma\_X^2\sigma\_Y^2}{\sigma\_X^2+\sigma\_Y^2}
> \end{equation}

Applying these to your problem, where $X=C$, $Y=\sqrt\beta\eta$, and $Z=z$ with the following statistics
\begin{align}
\mu\_X & = c(\theta,a)x+k \\
\mu\_Z & = c(\theta,a)x+k \\
\sigma\_X^2 & = \alpha \\
\sigma\_Y^2 & = \beta \\
\sigma\_{XZ} & = \sigma\_X^2 = \alpha \\
\sigma\_Z^2 & = \alpha +\beta
\end{align}
we thus obtain
\begin{align}
E(C\mid z) & = c(\theta,a)x+k+ \frac{\alpha}{\alpha+\beta}[z - (c(\theta,a)x+k)] \\
& = (c(\theta,a)x+k)\frac{\beta}{\alpha+\beta} + z\frac{\alpha}{\alpha+\beta}
\end{align}
and
\begin{align}
\text{Var}(C\mid z) & = \frac{\alpha\beta}{\alpha+\beta}
\end{align}


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