I can prove $X^{2}\approx 12$ through simulation in R, as shown in Appendix B. However, I do not know how to prove it mathmatically.

I can prove $X^{2}\approx 12$ through simulation in R, as shown in Appendix B. However, I do not know how to prove mathmatically:

$E(X^{2})=N$

A Pearson residual is defined as: $r_{i}(\theta)=\frac{y_{i}-E(y_{i}|\theta)}{\sqrt{Var(y_{i}|\theta)}}\tag{1}$ Sum of squared standard residuals $X^{2}$ is: $X^{2}=\sum_ir_i^2\tag{2}$ where $y_{i}$ is random variable. Also, {\theta} is random variable since it's about Bayesian, but this particular question is about "given $\theta$", so I think we can think $\theta$ not "random".

In a binomial model y[i] ~ dbin(theta, n[i]), it is said that for known $\theta$, we might expect $X^{2}$ to be around the number of y[i]. For an example in Appendix A, we can expect $X^{2}$ to be 12, given the known posterior $\theta$ = 0.12 (median value). The prior distribution of $\theta$ is uniform. $i$ has the value of 1, 2, 3, ..., 12 because there are twelve hospitals.

The problem is, I can prove it through simulation in R, as shown in Appendix B. However, I do not know how to prove it mathmatically.

A Pearson residual is defined as: $r_{i}(\theta)=\frac{y_{i}-E(y_{i}|\theta)}{\sqrt{Var(y_{i}|\theta)}}\tag{1}$ Sum of squared standard residuals $X^{2}$ is: $X^{2}=\sum_ir_i^2\tag{2}$ where $y_{i}$ is $i$th random variable ($i$ = 1, 2, ..., N). $\theta$ is also a random variable since it's about Bayesian inference, but this particular question is about "given $\theta$", so I think we can think $\theta$ not "random". N is the number of dependent variables y.

In a binomial model y[i] ~ dbin(theta, n[i]), it is said that

for known $\theta$, we might expect $X^{2}$ to be around N.

For an example in Appendix A, we can expect $X^{2}$ to be 12, given the known posterior $\theta$ = 0.12 (median value). The prior distribution of $\theta$ is uniform. $i$ has the value of 1, 2, 3, ..., 12 because there are twelve hospitals.

The problem is:

I can prove $X^{2}\approx 12$ through simulation in R, as shown in Appendix B. However, I do not know how to prove it mathmatically.

A Pearson residual is defined as: $r_{i}(\theta)=\frac{y_{i}-E(y_{i}|\theta)}{\sqrt{Var(y_{i}|\theta)}}\tag{1}$ Sum of squared standard residuals $X^{2}$ is: $X^{2}=\sum_ir_i^2\tag{2}$

In a binomial model y[i] ~ dbin(theta, n[i]), it is said that for known $\theta$, we might expect $X^{2}$ to be around the number of y[i]. For an example in Appendix A, we can expect $X^{2}$ to be 12, given the known posterior $\theta$ = 0.12.

A Pearson residual is defined as: $r_{i}(\theta)=\frac{y_{i}-E(y_{i}|\theta)}{\sqrt{Var(y_{i}|\theta)}}\tag{1}$ Sum of squared standard residuals $X^{2}$ is: $X^{2}=\sum_ir_i^2\tag{2}$ where $y_{i}$ is random variable. Also, {\theta} is random variable since it's about Bayesian, but this particular question is about "given $\theta$", so I think we can think $\theta$ not "random".

In a binomial model y[i] ~ dbin(theta, n[i]), it is said that for known $\theta$, we might expect $X^{2}$ to be around the number of y[i]. For an example in Appendix A, we can expect $X^{2}$ to be 12, given the known posterior $\theta$ = 0.12 (median value). The prior distribution of $\theta$ is uniform. $i$ has the value of 1, 2, 3, ..., 12 because there are twelve hospitals.