Expectation of correlated variables I'm looking to compare effects $ \delta = \frac{\mu_T - \mu_C}{\sigma}$ for two studies, compared with the same control group. In order to find the covariance of effects for treatment A and treatment B, I need to find
$$
E \left[ \left( \frac{\bar{A} - \bar{C}}{s} \right) \left( \frac{\bar{B} - \bar{C}}{s} \right) \right]
$$
I can assume $\bar{A}, \bar{B}$ and $\bar{C}$ are sample means and approximately normal, $s$ is pooled standard deviation with the usual relation to chi square. I believe all individual terms should be independent as well. 
My instinct is to simply expand 
$$
E \left[ \left( \bar{A} - \bar{C} \right) \left( \bar{B} - \bar{C} \right) (s^{-2}) \right]$$
$$E \left[ \bar{A} \bar{B} - \bar{A} \bar{C} - \bar{B} \bar{C} - \bar{C}^2 \right] E[(s^{-2})]
$$
But this loses the difference groupings that I'm interested in.
My advisor also said that this covariance should be related to the variance of the common components, but I'm failing to see how this is the case. 
Any advice on how to proceed would be very much appreciated.
 A: The covariance of two standardized mean difference computed with a common control group is given in:
Gleser, L. J., & Olkin, I. (2009). Stochastically dependent effect sizes. In H. Cooper, L. V. Hedges, & J. C. Valentine (Eds.), The handbook of research synthesis and meta-analysis (2nd ed., pp. 357–376). New York: Russell Sage Foundation.
Let's fix the notation a bit. Let $\bar{x}_1$ be the mean of the first group, $\bar{x}_2$ of the second group, and $\bar{x}_C$ of the control group. Let $s_1$, $s_2$, and $s_C$ denote the standard deviations of the scores within the respective groups, and $n_1$, $n_2$, and $n_C$ the respective group sizes. Finally, let $N = n_1 + n_2 + n_C$ denote the total sample size. Then
$$s_p = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2 + (n_C-1)s_C^2}{N - 3}}$$
is the pooled standard deviation of the three groups. Now the two standardized mean differences are:
$$d_1 = \frac{\bar{x}_1 - \bar{x}_C}{s_p}$$
and
$$d_2 = \frac{\bar{x}_2 - \bar{x}_C}{s_p}.$$
The variances of $d_1$ and $d_2$ can be estimated with:
$$Var(d_1) = \frac{1}{n_1} + \frac{1}{n_C} + \frac{d_1^2}{2N}$$
and
$$Var(d_2) = \frac{1}{n_2} + \frac{1}{n_C} + \frac{d_2^2}{2N}.$$
And finally, the covariance between $d_1$ and $d_2$ can be estimated with:
$$Cov(d_1, d_2) = \frac{1}{n_C} + \frac{d_1 d_2}{2N}.$$
