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This looks like a self-study question, so I will only give some hints: To study the distribution of a product of independent random variables, you can turn it into a problem of sums, by taking logs. …

If your income variable from the outset is categorical with values poor, vulnerable and non-poor, then after splitting the sample in three subsamples there is no variation left in that variable, so including … If you decide to include it, then the interpretation of the variable and its coefficiewnt will be different from in the full sample model: Now it differentiates by income within the "poor" group, say, …

If you want/need a new, different sample from the population the only way to get that is to sample from the population! … Its grounding, apart from its intuitive appeal, is in large sample theory, that is, approximations based on a large sample size. …

Then to simulate a sample of independent realization of $Y$, simulate first independent realizations of $X_1$, then independent from that, a sample of independent realizations from $X_2$ and then use $ …

Let $X=(X_1, \dotsc, X_n)^T$ which has covariance matrix $\Sigma=\sigma^2 R$ where the correlation matrix $R$ has diagonal elements $R_{ii}=1$ and off-diagonals $R_{ij}=a, i\not = j$ (here $-\frac1{n- …

If you are using a random sample of people, how can you choose how many from each group? … But maybe you want to estimate a model from this sample (and retrospectively obtained demographic data), to use in future ... Maybe in future you really want to find the people in group 3. …

The different subsamples come from interviewing a random sample of people, sampled independently from different locations, which are themselves a sample from the total population. … If that is zero, all the subpopulations have the same distribution, and we could really "treat the 10 subsamples like one large sample", otherwise we should not. …

In terms of your example, where $X$ is a person's weight, and there is a sample of $n$ persons, then $X_1$ is weight of first person in sample, $X_2$ is weight of second person in sample, and so on. …

No, the sample mean of an independent sample from a Poisson distribution is NOT Poisson distributed. … But, the sum of the sample ($n$ times the mean) do have a Poisson distribution, and all wanted probabilities can be calculated from that. …

The R code used: n <- 100 # "treatment" sample size m <- 100 # control sample size mu <- 10; sigma <- 3; delta <- 0.8 w <- 0.6 # fraction of treatment data really treated # Simulating some data: set.seed …

The general rule that people seem to use is to simply take $3/n$ (which in your case is 0.15%). This is sometimes called the "rule of three", and there are many questions here about this. (I do not …