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I need to find MLE estimator of $P(X \leq c)$ where X is $Normal(\theta,1)$, c is fixed. Note: $X_1,...,X_n$ are a random sample drawn from $Normal(\theta,1)$

Any hint on how to approach this problem? I know MLE is invariant and I thought of using the sample mean as my estimator of $\theta$ but I dont know how to find the MLE of $P(X \leq c)$.

Do I need to use the cdf or pdf? Also likelyhood is by definition the pdf at fixed $x_i$ that are drawn right? So I can't use the cdf?

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  • $\begingroup$ Please tag this as [self-study]. $\endgroup$ Commented May 14, 2018 at 9:36

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As I understand it, you have $X_1,\dots,X_n$ samples from $N(\theta, 1)$ distribution, where the mean of the distribution $\theta$ is unknown. You need to estimate $\Pr(X \le c)$.

The likelihood function is defined as

$$ L(\theta \mid X_1,\dots,X_n) = \prod_{i=1}^n f(X_i \mid \theta) $$

where $f$ is a probability density function (of $N(\theta, 1)$ distribution in your case). It is not defined in terms of cumulative distribution function, but to obtain your result the only thing you need to find is the unknown parameter $\theta$ and then just calculate the $\Pr(X \le c)$ using the cumulative distribution function of normal distribution!

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From the specified normal form $X_1, ..., X_n \sim \text{N}(\theta,1)$ you have $\mathbb{P}(X \leqslant c) = \Phi(X-c - \theta)$, which is a function of the unknown mean parameter $\theta$. The MLE for $\theta$ in this problem is $\hat{\theta} = \bar{X}$, and so, using the invariance property of the MLE we have:

$$\widehat{\mathbb{P}(X \leqslant c)} = \Phi(X-c - \hat{\theta}) = \Phi(X-c - \bar{x}).$$

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