Let X1, X2, …, Xn be a random sample from N(μ, σ^2). (a) If the constant b is defined by the equ

step 1 So, an important assumption here is that both the parameters mu and mu sigma square are unknown

Let X1, X2, …, Xn be a random sample from N(μ, σ^2). (a) If...

Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from $N\left(\mu, \sigma^{2}\right)$.
(a) If the constant $b$ is defined by the equation $\operatorname{Pr}(X \leq b)=0.90$, find the mle of $b$.
(b) If $c$ is given constant, find the mle of $\operatorname{Pr}(X \leq c)$.

Key Concepts

Maximum Likelihood Estimation (MLE)

MLE is a method for estimating the parameters of a statistical model by maximizing the likelihood function, which represents the probability of the observed data given the parameters. It provides point estimates that make the observed data most probable under the assumed model and is widely used due to its desirable asymptotic properties, such as consistency and efficiency.

Invariance Property of MLE

The invariance property of MLE states that if an estimator is obtained by maximizing the likelihood for a parameter, then any function of that parameter is estimated by applying the same function to the MLE of the parameter. This property simplifies the process of finding MLEs for functions of parameters, as the plug?in principle directly gives the estimator for the transformed parameter.

Quantile Function of the Normal Distribution

The quantile function of the normal distribution, which is the inverse of the cumulative distribution function (CDF), is used to determine the value below which a certain percentage of observations lie. In the context of parameter estimation, when a constant is defined via a probability statement such as Pr(X ? b) = q, the quantile function allows expressing b in terms of the parameters of the normal distribution. This concept is key to transforming probability statements into expressions involving the model parameters, which can then be estimated using MLE.

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