1. random variable

Let $\displaystyle \mu$ and $\displaystyle \sigma^{2}$ denote the mean and variance of the random variable X. Determine $\displaystyle E[\frac{(X-\mu)}{\sigma}]$ and $\displaystyle E{[\frac{(X-\mu)}{\sigma}]^{2}}.$Thank you for your help.

2. Hello,
Originally Posted by laser
Let $\displaystyle \mu$ and $\displaystyle \sigma^{2}$ denote the mean and variance of the random variable X. Determine $\displaystyle E[\frac{(X-\mu)}{\sigma}]$ and $\displaystyle E{[\frac{(X-\mu)}{\sigma}]^{2}}.$Thank you for your help.
Remember this : $\displaystyle \mathbb{E}(aX+b)=a\mathbb{E}(X)+b$

For the second one, by seeing your latex code, I guess you actually meant :
$\displaystyle \mathbb{E}\left(\left[\frac{(X-\mu)}{\sigma}\right]^2\right)$

Note that, by definition of the variance, it equals $\displaystyle \text{Var}\left(\frac{(X-\mu)}{\sigma}\right)+\left(\mathbb{E}\left(\frac{( X-\mu)}{\sigma}\right)\right)^2$

And remember that $\displaystyle \text{Var}(aX+b)=a^2 \text{Var}(X)$

You should know have all the necessary stuff for answering your questions
The text passage contains a mathematical question involving the expectation of a standardized random variable and its square. The passage also provides guidance and formulas to help solve the problem, but does not explicitly state the final answers.
single_question_with_answer

however, it's worth noting that while the passage provides the necessary steps and formulas to derive the answer, it does not explicitly state the final numerical or symbolic answers to the questions. therefore, it can be considered as providing an answer in the form of a solution method.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
1. random variable

Let $\displaystyle \mu$ and $\displaystyle \sigma^{2}$ denote the mean and variance of the random variable X. Determine $\displaystyle E[\frac{(X-\mu)}{\sigma}]$ and $\displaystyle E{[\frac{(X-\mu)}{\sigma}]^{2}}.$Thank you for your help.

2. Hello,
Originally Posted by laser
Let $\displaystyle \mu$ and $\displaystyle \sigma^{2}$ denote the mean and variance of the random variable X. Determine $\displaystyle E[\frac{(X-\mu)}{\sigma}]$ and $\displaystyle E{[\frac{(X-\mu)}{\sigma}]^{2}}.$Thank you for your help.
Remember this : $\displaystyle \mathbb{E}(aX+b)=a\mathbb{E}(X)+b$

For the second one, by seeing your latex code, I guess you actually meant :
$\displaystyle \mathbb{E}\left(\left[\frac{(X-\mu)}{\sigma}\right]^2\right)$

Note that, by definition of the variance, it equals $\displaystyle \text{Var}\left(\frac{(X-\mu)}{\sigma}\right)+\left(\mathbb{E}\left(\frac{( X-\mu)}{\sigma}\right)\right)^2$

And remember that $\displaystyle \text{Var}(aX+b)=a^2 \text{Var}(X)$

You should know have all the necessary stuff for answering your questions
user
The text passage contains a mathematical question involving the expectation of a standardized random variable and its square. The passage also provides guidance and formulas to help solve the problem, but does not explicitly state the final answers.

TYPE: SINGLE_QUESTION_WITH_ANSWER

However, it's worth noting that while the passage provides the necessary steps and formulas to derive the answer, it does not explicitly state the final numerical or symbolic answers to the questions. Therefore, it can be considered as providing an answer in the form of a solution method.