A population has a density function given by f(x)=(k+1)x^k 0&lt;x&lt;1 0 otherwise For n observations,x1,x2,x3...xn made from this population, find the maximum likelihood of k. Here's my attempt: I have got k=-1-n/(product of log xi fron i=1 to i=n). Im not sure if it's correct. Can somebody help?
Let $X_1, \cdots, X_n$ be a random sample from the PDF: $f(x;\theta) = (\theta + 1) x^{\theta}$ with $0&lt;x&lt;1$ and $\theta &gt; -1$.  The likelihood function is: \begin{align} L(\theta) &amp;= f(x_1, \cdots, x_n; \theta) \mathbb{1}\{0&lt;x&lt;1\} \mathbb{1}\{\theta &gt;-1\}\\ &amp;= \prod_{i=1}^{n}{f(x_i ; \theta)}\mathbb{1}\{0&lt;x&lt;1\} \mathbb{1}\{\theta &gt;-1\}\\ &amp;= \prod_{i=1}^{n}{(\theta+1)x_i^{\theta}}\mathbb{1}\{0&lt;x&lt;1\} \mathbb{1}\{\theta &gt;-1\} \\ &amp;=(\theta+1)^n \left( \prod_{i=1}^n{x_i}\right)^\theta \end{align}  We look at the log likelihood $l(\theta)$ and take the derivative of this since it is easier to deal with and it is allowed because the log is monotonic:  \begin{align} l(\theta) = n\log(\theta+1) + \theta \left( \sum_{i=1}^n{\log(x_i)} \right) \end{align}  $\implies \frac{d}{d\theta} = \frac{n}{\theta+1} + \sum_{i=1}^n{\log(x_i)} = 0$  $\implies \hat{\theta}_{MLE} = -\frac{n}{\sum_{i=1}^n{\log(x_i)}} - 1$  However, this doesn't look right.
Why do we use residuals to test the assumptions on errors in regression? Suppose that we have a model $Y_i = \beta_0 + \beta_1X_{i1} + \beta_2X_{i2} + \dots + \beta_kX_{ik} + \epsilon_i$.  Regression has a number of assumptions, such as that the errors $\epsilon_i$ should be normally distributed with mean zero and constant variance. I have been taught to check these assumptions using a normal QQ plot to test for normality of the residuals $e_i = Y_i - \hat{Y}_i$ and a residuals vs. fitted plot to check that the residuals vary around zero with constant variance.  However, these tests are all on the residuals, not the errors.  From what I understand, the errors are defined as the deviation of each observation from their 'true' mean value. So, we can write $\epsilon_i = Y_i - \mathbb{E}[Y_i]$. These errors cannot be observed by us. *  My question is this: how good of a job do the residuals do in mimicking the errors?   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