Above here is the information I've been given for one of my seminar questions, so far I have calculated the fisher information and from there I computed the asymptotic distribution for $\hat{\lambda}$ is:
$$\lambda_n = N\left(\lambda,\frac{1}{nI(\lambda)}\right) = N\left(\lambda, \frac{\lambda^2}{dn}\right)$$
After that I derived the 90% confidence interval for λ as:
$$T_{1,2}=\hat{\lambda} \pm \frac{z_{\alpha/2}}{\sqrt{nI(\lambda)}}= \frac{dn}{x} \pm 1.64 \frac{\lambda}{\sqrt{nd}}$$
And from here I need to find the exact 90% confidence interval but this is where I'm stuck. Can anyone provide any assistance?