While analyzing some parallel-computing related algorithm, I came across a probability distribution with a particularly nice property (at least to me), but I am unable to write it down explicitly.

Let $X$ be a r.v., let $F_{X}(s)$ be its p.g.f., and let $$B_{d}(s)=\sum_{i=0}^{d}\binom{d}{i}\left(\frac{1}{d}\right)^{i}\left(1-\frac{1}{d}\right)^{d-i}s^{i}=\left(1-\frac{1}{d}+\frac{s}{d}\right)^{d}$$ be a p.g.f. of $\left(d,\frac{1}{d}\right)$-binomial distributed r.v. $b_{d}$, for any integer $d>1$. The distribution in question may be defined as $$F_{X}\left(s\right)=s\sum_{i=0}^{d}\binom{d}{i}\left(\frac{1}{d}\right)^{i}\left(1-\frac{1}{d}\right)^{d-i}F_{X}^{i}\left(s\right)=s\left(1-\frac{1}{d}+\frac{F_{X}\left(s\right)}{d}\right)^{d}.$$

The r.v. $X$ may be intuitively described as follows: A student arrives into town with $1$, and earns an additional $b_{d}$ each month. At the end of each month, she also pays a rent of $1$. The r.v. $X$ basically measures the time before our student is thrown out of the apartment.

The title (as well as the student's future) may be a little dramatic, but is derived from the fact that if at some time the balance of our student is $k$, we expect her to stay for an additional $\sum_{i=1}^{k}X$ months (and this is independent of her past affairs). Also, the p.g.f. of $p$-geometric distribution $G_{p}$ is memoryless by the fact that

$$G_{p}(s)=s\cdot\text{Ber}_{p}\left(G_{p}\left(s\right)\right),$$

where $\text{Ber}_{p}$ is p.g.f. of $p$-Bernulli distributed r.v.. Similarly, in our case we can clearly write $F_{X}$ as

$$F_{X}\left(s\right)=s\cdot B_{d}\left(F_{X}\left(s\right)\right).$$

So far, I am only able to calculate (by derivation of $F_X$) that $\mathbb{E}(X)=\infty$, and also the first few coefficients of $F_X$, but I would be very interested in the precise description of $F_{X}$, either in a closed-form of p.g.f., or its coefficients, or cumulative distribution function of $X$.

While analyzing some parallel-computing related algorithm, I came across a probability distribution with a particularly nice property (at least to me), but I am unable to write it down explicitly.

Let $X$ be a r.v., let $F_{X}(s)$ be its p.g.f., and let $$B_{d}(s)=\sum_{i=0}^{d}\binom{d}{i}\left(\frac{1}{d}\right)^{i}\left(1-\frac{1}{d}\right)^{d-i}s^{i}=\left(1-\frac{1}{d}+\frac{s}{d}\right)^{d}$$ be a p.g.f. of $\left(d,\frac{1}{d}\right)$-binomial distributed r.v. $b_{d}$, for any integer $d>1$. The distribution in question may be defined as $$F_{X}\left(s\right)=s\sum_{i=0}^{d}\binom{d}{i}\left(\frac{1}{d}\right)^{i}\left(1-\frac{1}{d}\right)^{d-i}F_{X}^{i}\left(s\right)=s\left(1-\frac{1}{d}+\frac{F_{X}\left(s\right)}{d}\right)^{d}.$$

The r.v. $X$ may be intuitively described as follows: A student arrives into town with $1$, and earns an additional $b_{d}$ each month. At the end of each month, she also pays a rent of $1$. The r.v. $X$ basically measures the time before our student is thrown out of the apartment.

If at some time the balance of our student is $k$, we expect her to stay for an additional $\sum_{i=1}^{k}X$ months (and this is independent of her past affairs). Also, the p.g.f. of $p$-geometric distribution $G_{p}$ is memoryless by the fact that

$$G_{p}(s)=s\cdot\text{Ber}_{p}\left(G_{p}\left(s\right)\right),$$

where $\text{Ber}_{p}$ is p.g.f. of $p$-Bernulli distributed r.v.. Similarly, in our case we can clearly write $F_{X}$ as

$$F_{X}\left(s\right)=s\cdot B_{d}\left(F_{X}\left(s\right)\right).$$

So far, I am only able to calculate (by derivation of $F_X$) that $\mathbb{E}(X)=\infty$, and also the first few coefficients of $F_X$, but I would be very interested in the precise description of $F_{X}$, either in a closed-form of p.g.f., or its coefficients, or cumulative distribution function of $X$.

Dear MathOverflow community,

While analyzing some parallel-computing related algorithm, I came across a probability distribution with a particularly nice property (at least to me), but I am unable to write it down explicitly.

Let $X$ be a r.v., let $F_{X}(s)$ be its p.g.f., and let $$B_{d}(s)=\sum_{i=0}^{d}\binom{d}{i}\left(\frac{1}{d}\right)^{i}\left(1-\frac{1}{d}\right)^{d-i}s^{i}=\left(1-\frac{1}{d}+\frac{s}{d}\right)^{d}$$ be a p.g.f. of $\left(d,\frac{1}{d}\right)$-binomial distributed r.v. $b_{d}$, for any integer $d>1$. The distribution in question may be defined as $$F_{X}\left(s\right)=s\sum_{i=0}^{d}\binom{d}{i}\left(\frac{1}{d}\right)^{i}\left(1-\frac{1}{d}\right)^{d-i}F_{X}^{i}\left(s\right)=s\left(1-\frac{1}{d}+\frac{F_{X}\left(s\right)}{d}\right)^{d}.$$

The r.v. $X$ may be intuitively described as follows: A student arrives into town with $1$, and earns an additional $b_{d}$ each month. At the end of each month, she also pays a rent of $1$. The r.v. $X$ basically measures the time before our student is thrown out of the apartment.

The title (as well as the student's future) may be a little dramatic, but is derived from the fact that if at some time the balance of our student is $k$, we expect her to stay for an additional $\sum_{i=1}^{k}X$ months (and this is independent of her past affairs). Also, the p.g.f. of $p$-geometric distribution $G_{p}$ is memoryless by the fact that

$$G_{p}(s)=s\cdot\text{Ber}_{p}\left(G_{p}\left(s\right)\right),$$

where $\text{Ber}_{p}$ is p.g.f. of $p$-Bernulli distributed r.v.. Similarly, in our case we can clearly write $F_{X}$ as

$$F_{X}\left(s\right)=s\cdot B_{d}\left(F_{X}\left(s\right)\right).$$

So far, I am only able to calculate (by derivation of $F_X$) that $\mathbb{E}(X)=\infty$, and also the first few coefficients of $F_X$, but I would be very interested in the precise description of $F_{X}$, either in a closed-form of p.g.f., or its coefficients, or cumulative distribution function of $X$.

While analyzing some parallel-computing related algorithm, I came across a probability distribution with a particularly nice property (at least to me), but I am unable to write it down explicitly.

Let $X$ be a r.v., let $F_{X}(s)$ be its p.g.f., and let $$B_{d}(s)=\sum_{i=0}^{d}\binom{d}{i}\left(\frac{1}{d}\right)^{i}\left(1-\frac{1}{d}\right)^{d-i}s^{i}=\left(1-\frac{1}{d}+\frac{s}{d}\right)^{d}$$ be a p.g.f. of $\left(d,\frac{1}{d}\right)$-binomial distributed r.v. $b_{d}$, for any integer $d>1$. The distribution in question may be defined as $$F_{X}\left(s\right)=s\sum_{i=0}^{d}\binom{d}{i}\left(\frac{1}{d}\right)^{i}\left(1-\frac{1}{d}\right)^{d-i}F_{X}^{i}\left(s\right)=s\left(1-\frac{1}{d}+\frac{F_{X}\left(s\right)}{d}\right)^{d}.$$

The r.v. $X$ may be intuitively described as follows: A student arrives into town with $1$, and earns an additional $b_{d}$ each month. At the end of each month, she also pays a rent of $1$. The r.v. $X$ basically measures the time before our student is thrown out of the apartment.

The title (as well as the student's future) may be a little dramatic, but is derived from the fact that if at some time the balance of our student is $k$, we expect her to stay for an additional $\sum_{i=1}^{k}X$ months (and this is independent of her past affairs). Also, the p.g.f. of $p$-geometric distribution $G_{p}$ is memoryless by the fact that

$$G_{p}(s)=s\cdot\text{Ber}_{p}\left(G_{p}\left(s\right)\right),$$

where $\text{Ber}_{p}$ is p.g.f. of $p$-Bernulli distributed r.v.. Similarly, in our case we can clearly write $F_{X}$ as

$$F_{X}\left(s\right)=s\cdot B_{d}\left(F_{X}\left(s\right)\right).$$

So far, I am only able to calculate (by derivation of $F_X$) that $\mathbb{E}(X)=\infty$, and also the first few coefficients of $F_X$, but I would be very interested in the precise description of $F_{X}$, either in a closed-form of p.g.f., or its coefficients, or cumulative distribution function of $X$.

Dear MathOverflow community,

While analyzing some parallel-computing related algorithm, I came across a probability distribution with a particularly nice property (at least to me), but I am unable to write it down explicitly.

Let $X$ be a r.v., let $F_{X}(s)$ be its p.g.f., and let $$B_{d}(s)=\sum_{i=0}^{d}\binom{d}{i}\left(\frac{1}{d}\right)^{i}\left(1-\frac{1}{d}\right)^{d-i}s^{i}=\left(1-\frac{1}{d}+\frac{s}{d}\right)^{d}$$ be a p.g.f. of $\left(d,\frac{1}{d}\right)$-binomial distributed r.v. $b_{d}$, for any integer $d>1$. The distribution in question may be defined as $$F_{X}\left(s\right)=s\sum_{i=0}^{d}\binom{d}{i}\left(\frac{1}{d}\right)^{i}\left(1-\frac{1}{d}\right)^{d-i}F_{X}^{i}\left(s\right)=s\left(1-\frac{1}{d}+\frac{F_{X}\left(s\right)}{d}\right)^{d}.$$

The r.v. $X$ may be intuitively described as follows: A student arrives into town with $1$, and earns an additional $b_{d}$ each month. At the end of each month, she also pays a rent of $1$. The r.v. $X$ basically measures the time before our student is thrown out of the apartment.

The title (as well as the student's future) may be a little dramatic, but is derived from the fact that if at some time the balance of our student is $k$, we expect her to stay for an additional $\sum_{i=1}^{k}X$ months (and this is independent of her past affairs). Also, the p.g.f. of $p$-geometric distribution $G_{p}$ is memoryless by the fact that

$$G_{p}(s)=s\cdot\text{Ber}_{p}\left(G_{p}\left(s\right)\right),$$

where $\text{Ber}_{p}$ is p.g.f. of $p$-Bernulli distributed r.v.. Similarly, in our case we can clearly write $F_{X}$ as

$$F_{X}\left(s\right)=s\cdot B_{d}\left(F_{X}\left(s\right)\right).$$

So far, I am only able to calculate the first few coefficients, but I would be very interested in the precise description of $F_{X}$, either in a closed-form of p.g.f., or its coefficients, or cumulative distribution function of $X$.

Dear MathOverflow community,

While analyzing some parallel-computing related algorithm, I came across a probability distribution with a particularly nice property (at least to me), but I am unable to write it down explicitly.

Let $X$ be a r.v., let $F_{X}(s)$ be its p.g.f., and let $$B_{d}(s)=\sum_{i=0}^{d}\binom{d}{i}\left(\frac{1}{d}\right)^{i}\left(1-\frac{1}{d}\right)^{d-i}s^{i}=\left(1-\frac{1}{d}+\frac{s}{d}\right)^{d}$$ be a p.g.f. of $\left(d,\frac{1}{d}\right)$-binomial distributed r.v. $b_{d}$, for any integer $d>1$. The distribution in question may be defined as $$F_{X}\left(s\right)=s\sum_{i=0}^{d}\binom{d}{i}\left(\frac{1}{d}\right)^{i}\left(1-\frac{1}{d}\right)^{d-i}F_{X}^{i}\left(s\right)=s\left(1-\frac{1}{d}+\frac{F_{X}\left(s\right)}{d}\right)^{d}.$$

The r.v. $X$ may be intuitively described as follows: A student arrives into town with $1$, and earns an additional $b_{d}$ each month. At the end of each month, she also pays a rent of $1$. The r.v. $X$ basically measures the time before our student is thrown out of the apartment.

The title (as well as the student's future) may be a little dramatic, but is derived from the fact that if at some time the balance of our student is $k$, we expect her to stay for an additional $\sum_{i=1}^{k}X$ months (and this is independent of her past affairs). Also, the p.g.f. of $p$-geometric distribution $G_{p}$ is memoryless by the fact that

$$G_{p}(s)=s\cdot\text{Ber}_{p}\left(G_{p}\left(s\right)\right),$$

where $\text{Ber}_{p}$ is p.g.f. of $p$-Bernulli distributed r.v.. Similarly, in our case we can clearly write $F_{X}$ as

$$F_{X}\left(s\right)=s\cdot B_{d}\left(F_{X}\left(s\right)\right).$$

So far, I am only able to calculate (by derivation of $F_X$) that $\mathbb{E}(X)=\infty$, and also the first few coefficients of $F_X$, but I would be very interested in the precise description of $F_{X}$, either in a closed-form of p.g.f., or its coefficients, or cumulative distribution function of $X$.