The Cramér-Rao Lower Bound (CRLB) provides a lower bound on the variance of unbiased estimators of a parameter. To find the CRLB for the variance of an unbiased estimator of the parameter \(\theta\), we need to calculate the Fisher information \(I(\theta)\) for a single observation from the given probability density function (pdf) \(f(x;\theta)\). The Fisher information is given by: \[ I(\theta) = -E\left[\frac{\partial^2}{\partial \theta^2} \ln f(X;\theta)\right] \] Given the pdf: \[ f(x;\theta) = \begin{cases} 3a\theta^2 e^{-\theta x^3}, & 0 < x < \infty \\ 0, & \text{otherwise} \end{cases} \] First, we need to compute the natural logarithm of the pdf: \[ \ln f(x;\theta) = \ln(3a) + 2\ln(\theta) - \theta x^3 \] Now, we take the first derivative with respect to \(\theta\): \[ \frac{\partial}{\partial \theta} \ln f(x;\theta) = \frac{2}{\theta} - x^3 \] Next, we take the second derivative with respect to \(\theta\): \[ \frac{\partial^2}{\partial \theta^2} \ln f(x;\theta) = -\frac{2}{\theta^2} \] Now, we calculate the expected value of the second derivative. Since the second derivative does not depend on \(x\), the expected value is simply the negative of the second derivative itself: \[ E\left[-\frac{\partial^2}{\partial \theta^2} \ln f(X;\theta)\right] = E\left[\frac{2}{\theta^2}\right] = \frac{2}{\theta^2} \] Therefore, the Fisher information for a single observation is: \[ I(\theta) = \frac{2}{\theta^2} \] For a sample of size \(n\), the Fisher information is \(n\) times the information for a single observation, since the observations are independent: \[ I_n(\theta) = nI(\theta) = \frac{2n}{\theta^2} \] The Cramér-Rao Lower Bound for the variance of any unbiased estimator \(\hat{\theta}\) of \(\theta\) is then given by the reciprocal of the Fisher information for the sample: \[ \text{Var}(\hat{\theta}) \geq \frac{1}{I_n(\theta)} = \frac{\theta^2}{2n} \] This is the Cramér-Rao Lower Bound for the variance of unbiased estimators of the parameter \(\theta\).

The Cramér-Rao Lower Bound (CRLB) provides a lower bound on the variance of unbiased estimators of a parameter. To find the CRLB for the variance of an unbiased estimator of the parameter $\theta$, we need to calculate the Fisher information $I(\theta)$ for a single observation from the given probability density function (pdf) $f(x;\theta)$. The Fisher information is given by: $I(\theta) = -E\left[\frac{\partial^2}{\partial \theta^2} \ln f(X;\theta)\right]$ Given the pdf: $f(x;\theta) = \begin{cases} 3a\theta^2 e^{-\theta x^3}, & 0 < x < \infty \\ 0, & \text{otherwise} \end{cases}$ First, we need to compute the natural logarithm of the pdf: $\ln f(x;\theta) = \ln(3a) + 2\ln(\theta) - \theta x^3$ Now, we take the first derivative with respect to $\theta$: $\frac{\partial}{\partial \theta} \ln f(x;\theta) = \frac{2}{\theta} - x^3$ Next, we take the second derivative with respect to $\theta$: $\frac{\partial^2}{\partial \theta^2} \ln f(x;\theta) = -\frac{2}{\theta^2}$ Now, we calculate the expected value of the second derivative. Since the second derivative does not depend on $x$, the expected value is simply the negative of the second derivative itself: $E\left[-\frac{\partial^2}{\partial \theta^2} \ln f(X;\theta)\right] = E\left[\frac{2}{\theta^2}\right] = \frac{2}{\theta^2}$ Therefore, the Fisher information for a single observation is: $I(\theta) = \frac{2}{\theta^2}$ For a sample of size $n$, the Fisher information is $n$ times the information for a single observation, since the observations are independent: $I_n(\theta) = nI(\theta) = \frac{2n}{\theta^2}$ The Cramér-Rao Lower Bound for the variance of any unbiased estimator $\hat{\theta}$ of $\theta$ is then given by the reciprocal of the Fisher information for the sample: $\text{Var}(\hat{\theta}) \geq \frac{1}{I_n(\theta)} = \frac{\theta^2}{2n}$ This is the Cramér-Rao Lower Bound for the variance of unbiased estimators of the parameter $\theta$.