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Let $X_1, X_2,\dots,X_n$ be $n$ i.i.d. geometric distributed random variables with successful probability $p$, that is $$P(X_i=k)=(1-p)^{k-1}p$$

Let $Y_i=X_i-\mathbb{E}(X_i) = X_i-\frac{1}{p}$, so $Y_1,\dots,Y_n$ are also i.i.d. random variables.

Let $S_k=Y_1+\dots+Y_k$, and $S_k^+=\max(0,S_k)$

I want to know a close form of $\mathbb{E}(S_k^+)$

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2 Answers 2

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Let $X$ denote a discrete random variable with finite mean $\mu$. From $$E[X-\mu] = 0 = \sum_i (u_i - \mu) p_X(u_i) = \sum_{i\colon u_i > \mu} (u_i - \mu) p_X(u_i) + \sum_{i\colon u_i < \mu} (u_i - \mu) p_X(u_i),$$ we get that $$\sum_{i\colon u_i > \mu} (u_i - \mu)p_X(u_i) = \sum_{i\colon u_i < \mu} (\mu - u_i) p_X(u_i) $$ and so if $Y = \max\{0, X-\mu\}$, we have that $$\begin{align*} E[Y] &= \sum_i \max\{0, u_i - \mu\} p_X(u_i)\\ &= \sum_{i\colon u_i > \mu} (u_i - \mu)p_X(u_i)\\ &= \sum_{i\colon u_i < \mu} (\mu - u_i) p_X(u_i). \end{align*}$$ In the question under consideration, $X = \sum_{j=1}^k X_j$ is a negative binomial random variable with parameters $(k,p)$ and mean $\frac{k}{p}$. Its pmf is given by $$p_X(n) = \binom{n - 1}{k-1} p^k (1 - p)^{n-k}, ~~ n = k, k+1, k+2, \ldots $$ Hence, $$\begin{align*} E[S_k^+] &= E\left[\max\left\{0, X-\frac{k}{p}\right\}\right]\\ &= \sum_{n = \lceil \frac{k}{p}\rceil}^\infty \left(n - \frac{k}{p}\right)\binom{n - 1}{k-1} p^k (1 - p)^{n-k}\\ &= \sum_{n = k}^{\lfloor \frac{k}{p}\rfloor} \left(\frac{k}{p}- n \right)\binom{n - 1}{k-1} p^k (1 - p)^{n-k}. \end{align*}$$ If $k/p$ is an integer, one of the terms in each sum is $0$. The first form of the answer has already been given by @Alen but the second might be easier to evaluate numerically since it is a finite sum.

Neither form gives the floor of the expected value as has been alleged by OP Fan Zhang in comments on the main question.

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Since $\sum\limits_{i = 1}^k {{X_i}} $ has negative binomial distribution with parameters $k$ and $1-p$, we have $$ \mathbb{E}\left[ {S_k^ + } \right] = \sum\limits_{j = \left\lceil {\frac{k} {p}} \right\rceil }^\infty {\left( {j - \frac{k} {p}} \right)\left( \begin{gathered} j + k - 1 \\ j \\ \end{gathered} \right){p^k}{{\left( {1 - p} \right)}^j}} $$

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  • $\begingroup$ This disregards jensen's inequality, and you can calculate it easier/to a number. $\endgroup$ Jan 8, 2012 at 14:57
  • $\begingroup$ ...or not to the 'easier' part. My bad. It still seems to disregard jensen's inequality. $\endgroup$ Jan 8, 2012 at 15:09
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    $\begingroup$ @gnometorule: What are you talking about? $\endgroup$
    – Did
    Jan 8, 2012 at 15:14
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    $\begingroup$ Alen: Added a link. $\endgroup$
    – Did
    Jan 8, 2012 at 15:17
  • $\begingroup$ Is $\mathbb{E}\left[ {S_k^ + } \right] = \mathbb{E}[S_k^+]$? $\endgroup$
    – Fan Zhang
    Jan 8, 2012 at 16:22

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