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Copy file name to clipboardExpand all lines: lectures/hoist_failure.md
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@@ -124,7 +124,9 @@ In particular, if $y_1$ is log normal with parameters $(\mu_1, \sigma_1^2)$ and
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$y_2$ is log normal with parameters $(\mu_2, \sigma_2^2)$, then the product $y_1 y_2$ is log normal
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with parameters $(\mu_1 + \mu_2, \sigma_1^2 + \sigma_2^2)$.
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**Note:** While the product of two log normal distributions is log normal, the **sum** of two log normal distributions is **not** log normal.
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```{note}
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While the product of two log normal distributions is log normal, the **sum** of two log normal distributions is **not** log normal.
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```
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This observation sets the stage for challenge that confronts us in this lecture, namely, to approximate probability distributions of **sums** of independent log normal random variables.
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<!-- #region -->
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Now we shall set a grid length $I$ and a grid increment size $m =1$ for our discretizations.
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**Note**: We set $I$ equal to a power of two because we want to be free to use a Fast Fourier Transform
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```{note}
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We set $I$ equal to a power of two because we want to be free to use a Fast Fourier Transform
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to compute a convolution of two sequences (discrete distributions).
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```
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We recommend experimenting with different values of the power $p$ of 2.
**Note:** Because the failure rates are all very small, log normal distributions with the
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```{note}
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Because the failure rates are all very small, log normal distributions with the
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above parameter values actually describe $P(A_i)$ times $10^{-09}$.
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```
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So the probabilities that we'll put on the $x$ axis of the probability mass function and associated cumulative distribution function should be multiplied by $10^{-09}$
Copy file name to clipboardExpand all lines: lectures/house_auction.md
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@@ -43,9 +43,10 @@ We describe two distinct mechanisms
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* A special case of a Groves-Clarke {cite}`Groves_73`, {cite}`Clarke_71` mechanism with a benevolent social planner
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```{note}
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In 1994, the multiple rounds, ascending bid auction was actually used by Stanford University to sell leases to 9 lots on the Stanford campus to eligible faculty members.
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```
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**Note:** In 1994, the multiple rounds, ascending bid auction was actually used by Stanford University to sell leases to 9 lots on the Stanford campus to eligible faculty members.
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