# Correlated Bernoulli Trials

Suppose there are $$n$$ dependent Bernoulli trials, $$X_{1}$$,...,$$X_{n}$$ with $$% X_{j}\in \{1,0\}$$ and $$\Pr (X_{j}=1)=q$$ for all $$j=1,...,n$$. For any $$% n\geqslant 2$$ dependent Bernoulli trials, in the most general case, defining a joint distribution of $$\{X_{j}\}_{j=1}^{n}$$ requires specification of $$2^{n}-1$$ parameters. As in Witt (2014), I make two assumptions to reduce the required parameter set to two: $$(q,\rho )$$, where, $$q$$ is already defined and $$\rho$$ is defined in Assumption 2.

ASSUMPTION 1: $$(X_{1},\ldots ,X_{n})$$ are exchangeable, that is, $$\begin{equation*} \Pr (X_{1}=\lambda _{1},\ldots ,X_{n}=\lambda _{n})=\Pr (X_{\pi (1)}=\lambda _{1},\ldots ,X_{\pi (n)}=\lambda _{n}), \end{equation*}$$ where $$\lambda _{1},\ldots ,\lambda _{n}\in \{0,1\}$$ and $$\pi$$ is a permutation of $$\{1,\ldots ,n\}.$$

ASSUMPTION 2: Constant pairwise correlation, that is, $$\begin{equation*} \rho =\operatorname{Corr}(X_{i},X_{j}|X_{\pi (1)}=1,\ldots ,X_{\pi (k)}=1)>0 \end{equation*}$$ for $$k=0,\ldots ,n-2$$ and for any permutation $$\pi (1),\ldots ,\pi (k)$$ of $$k$$ terms of the sequence of $$n-2$$ integers $$1,\ldots ,n$$ excluding $$i$$ and $$j$$

With these two assumptions, Witt (2014) shows $$$$\Pr (X_{j}=1|X_{\pi (1)}=1,\ldots ,X_{\pi (j-1)}=1)=1-\left( 1-q\right) \left( 1-\rho \right) ^{j-1}.$$$$

QUESTION: To show that $$\begin{equation*} \frac{\partial }{\partial \rho }\frac{\Pr (X_{k+1}=0,...,X_{n}=0|X_{1}=1,\ldots ,X_{k}=1)}{\Pr (X_{k+1}=0,...,X_{n}=0|X_{1}=0,\ldots ,X_{k}=0)}<0. \end{equation*}$$

My strategy is to show that $$\frac{\partial }{\partial \rho }\Pr (X_{k+1}=0,...,X_{n}=0|X_{1}=1,\ldots ,X_{k}=1)<0$$ and $$\frac{\partial }{% \partial \rho }\Pr (X_{k+1}=0,...,X_{n}=0|X_{1}=0,\ldots ,X_{k}=0)>0$$. Intuitively it makes sense that when Bernoulli trials are positively correlated, as the correlation increases, the probability that the next $$n-k$$ trials will be failures given that the first $$k$$ trials are successes should decrease. Conversely, as correlation increases, the probability that the next $$n-k$$ trials will be failures given that the first $$k$$ trials are also failures should increase. However, how do I prove it mathematically?

I can prove that $$\frac{\partial }{\partial \rho }\Pr (X_{k+1}=0|X_{1}=1,\ldots ,X_{k}=1)<0$$, by showing that $$\Pr (X_{k+1}=0|X_{1}=1,\ldots ,X_{k}=1)= \left( 1-q\right) \left( 1-\rho \right) ^{k}$$, which decreases with $$\rho$$.

But I am getting stuck at proving $$\frac{\partial }{\partial \rho }\Pr (X_{k+1}=0,X_{k+2}=0|X_{1}=1,\ldots ,X_{k}=1)<0.$$

I will appreciate any help.

[Please note that I have checked all related threads, but could not find any useful help].

References:

• +1 But did you notice that "$p$" and "$\rho$" are rendered almost identically? Ouch! – whuber Feb 13 '19 at 13:40