Question
Problem 3. The sample mean points possible (graded)Let X be a continuous random variable We know that it takes values between 0 and 6, but we do not know its distribution or its mean and variance; although we know that its variance is at most 4. We are interested in estimating the mean of X , which we denote by h. To estimate h, we take n ii.d. samples X1, Xz, which all have the same distribution as X, and compute the sample meanH =iEx
Problem 3. The sample mean points possible (graded) Let X be a continuous random variable We know that it takes values between 0 and 6, but we do not know its distribution or its mean and variance; although we know that its variance is at most 4. We are interested in estimating the mean of X , which we denote by h. To estimate h, we take n ii.d. samples X1, Xz, which all have the same distribution as X, and compute the sample mean H = iEx
Answers
Suppose that the random variable $X$ has the continuous uniform distribution $$f(x)=\left\{\begin{array}{ll}1, & 0 \leq x \leq 1 \\0, & \text { otherwise }\end{array}\right.$$ Suppose that a random sample of $n=12$ observations is selected from this distribution. What is the approximate probability distribution of $\bar{X}-6 ?$ Find the mean and variance of this quantity.
It is important to get a solid foundation of the vocabulary for sampling distributions. For us to start this next part. Now, a sampling distribution is different than a sample distribution. So when we're talking about a sampling distribution, you're repeatedly taking samples out of a population with a specific size end and you're calculating some statistic about that sample. And then you're actually looking at the distribution of that statistic and how we can apply that later in our inferential statistics. So our first thing with vocabulary is that if we're looking at the probability distribution of all the possible values of the random variable X. Bar. Remember ex far as a sample mean? So I grab a sample and I find the mean of that sample. And then I grab another sample size, same size end, and I find the mean of that sample and I do that over and over and over. And then I look at the probability distribution of those um sample mean values. And that expert is calculated from a sample of size and from a population whose mean is new and whose standard deviation is sigma. That process produces a probability distribution known as the sampling distribution of exper.
Well here we are, given a random variable X. That has a discreet uniform distribution On the into jersey fixed between one and 3 and first for us to determine the mean and the variance. So the mean and the various formulas are given here. So the meanest P plus K over two, which is basically wants to to experience, is the minus a plus one squared minus 1/12. So three squared is nine, 9 -1 is eight. 8/12, yes .667. So we have our mean and variance.
For this exercise, we have a random variable X. That has a discreet uniform distribution on the integers one through three. So in other words, the probability of X taking on either of the three values 12 or three is one third for each. We are asked in the question to determine the mean and the variance of X. The mean or expected value of X. When it is a uniform discrete variable is A plus B, divided by two where a is the smallest integer, and B is the largest integer, And that gives us a mean of two. Now to solve the variants, we use this formula, so we have three minus one plus one, All squared -1 all over 12. And this comes out to you to over three. And so this random variable with a discreet uniform distribution has a mean of two and a variance of 2/3.