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 Okay, in this question for the other part we have as x is approximately n mu zigma square. Hence

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 $P(X \leq b)=0.90$, find the mle of $b$.
(b) If $c$ is given constant, find the mle of $P(X \leq c)$.

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. In the context of the normal distribution, the MLEs for the mean and variance are the sample mean and sample variance (with appropriate divisor for unbiasedness), and these estimates are used as plug-ins in subsequent inference tasks.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its mean (?) and variance (?²). It has a bell-shaped density function, and many properties, such as the distribution of standardized variables and the role of quantiles, are well understood. The fact that linear transformations of normally distributed variables remain normal is often used in inference.

Quantiles of the Distribution

Quantiles are values that partition the distribution into intervals with given probabilities. For a normal distribution, a quantile corresponding to a cumulative probability (like 0.90) can be expressed as ? plus a multiple of ?. The value from the standard normal table (e.g., z_0.90) is used to compute the quantile for any normal random variable.

Plug-in Principle

The plug-in principle is the approach of substituting unknown parameters with their estimates when calculating functions of these parameters. In this context, when calculating probabilities or quantiles that depend on ? and ?, the MLEs (i.e., the sample mean and sample standard deviation) are plugged into the expression, yielding the MLE of the desired function.

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