<p>Here <span class="math-container">$F$</span> is the CDF of non-negative random variable <span class="math-container">$X$</span> with support <span class="math-container">$[0,\omega]$</span>, and <span class="math-container">$f$</span> denotes its PDF.
<span class="math-container">\begin{eqnarray*}\mathbb{E}\left(\frac{1}{\lambda(X)}\right) &amp; = &amp; \int_0^\omega\left(\frac{1-F(x)}{f(x)}\right)f(x) dx \\ &amp; = &amp; \int_0^\omega (1-F(x))dx \\ &amp; = &amp; \int_0^\omega \int_x^\omega f(t) dt dx \\ &amp; = &amp; \int_0^\omega\int_0^tf(t) dxdt \\ &amp;=&amp; \int_0^\omega f(t)\int_0^t dx dt \\ &amp;=&amp; \int_0^\omega f(t)t dt \\ &amp; = &amp; \mathbb{E}(X)\end{eqnarray*}</span></p>

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2024-08-07T19:07:54.943Z
2024-08-07T20:07:42.127Z
2024-08-07T20:07:42.127Z
Amit
<p>I am reading Krishna's Auction Theory. I am currently stuck on Appendix A where he writes,
<a href="https://i.sstatic.net/6qkGJqBM.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/6qkGJqBM.png" alt="enter image description here" /></a></p>
<p>Now I understand equation A2,</p>
<p>what I don't follow is that for a hazard rate defined as below:
<a href="https://i.sstatic.net/6HVpUrLB.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/6HVpUrLB.png" alt="enter image description here" /></a></p>
<p>he writes
<a href="https://i.sstatic.net/yrdv31V0.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/yrdv31V0.png" alt="enter image description here" /></a></p>
<p>I have tried for a long time but unable to get this.</p>

<statistics><auctions><mechanism-design><probability>
Expected Value of random variable in terms of hazard function
Karan Kumar
# Expected Value of random variable in terms of hazard function

(Asked by: username_0 on 2024-08-07)

I am reading Krishna's Auction Theory. I am currently stuck on Appendix A where he writes,
[![enter image description here](https://i.sstatic.net/6qkGJqBM.png)](https://i.sstatic.net/6qkGJqBM.png)

Now I understand equation A2,

what I don't follow is that for a hazard rate defined as below:
[![enter image description here](https://i.sstatic.net/6HVpUrLB.png)](https://i.sstatic.net/6HVpUrLB.png)

he writes
[![enter image description here](https://i.sstatic.net/yrdv31V0.png)](https://i.sstatic.net/yrdv31V0.png)

I have tried for a long time but unable to get this.

**Tags:** statistics, auctions, mechanism-design, probability

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## 1 answers

### Answer by username_1 on 2024-08-07. Score: 3 | Accepted answer

Here $F$ is the CDF of non-negative random variable $X$ with support $[0,\omega]$, and $f$ denotes its PDF.
\begin{eqnarray\*}\mathbb{E}\left(\frac{1}{\lambda(X)}\right) & = & \int\_0^\omega\left(\frac{1-F(x)}{f(x)}\right)f(x) dx \\ & = & \int\_0^\omega (1-F(x))dx \\ & = & \int\_0^\omega \int\_x^\omega f(t) dt dx \\ & = & \int\_0^\omega\int\_0^tf(t) dxdt \\ &=& \int\_0^\omega f(t)\int\_0^t dx dt \\ &=& \int\_0^\omega f(t)t dt \\ & = & \mathbb{E}(X)\end{eqnarray\*}


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