(Solved): 1 Sampling Distributions \\& CLT Problem 1. Suppose \\( Y_{1}, Y_{2}, Y_{3} \\) are random samples ...

1 Sampling Distributions \\& CLT Problem 1. Suppose \\( Y_{1}, Y_{2}, Y_{3} \\) are random samples from \\( N(1,1) \\). Let, \\[ \\bar{Y}=\\frac{Y_{1}+Y_{2}+Y_{3}}{3} \\& \\quad S^{2}=\\frac{1}{2} \\sum_{i=1}^{3}\\left(Y_{i}-\\bar{Y}\\right)^{2} . \\] What's the distribution of \\( U=2 S^{2} \\) ? Using the distribution of \\( U \\), compute \\( E\\left[S^{2}\\right] \\) and \\( \\operatorname{Var}\\left[S^{2}\\right] \\). Problem 2. Let \\( Y_{1}, Y_{2}, \\ldots, Y_{n} \\) be a random sample from \\( N\\left(\\mu, \\sigma^{2}\\right) \\). Define, \\[ \\bar{Y}=\\frac{1}{n} \\sum_{i=1}^{n} Y_{i} \\quad \\& \\quad S^{2}=\\frac{1}{n-1} \\sum_{i=1}^{n}\\left(Y_{i}-\\bar{Y}\\right)^{2} . \\] i) Suppose that \\( \\sigma^{2}=9 \\) and it is desired to have a sample mean \\( Y \\) within 1.5 unites of the population mean \\( \\mu \\) with probability 0.95 What is the sample size \\( n \\) required to achieve this? ii) Suppose that the variance \\( \\sigma^{2} \\) is unknown and it is desired to have the sample mean within \\( \\frac{1.74}{\\sqrt{n}} S \\) units of the population mean \\( \\mu \\) with probability 0.9 . What is the sample size \\( n \\) required to achieve this? Problem 3. Let \\( Y_{1}, Y_{2}, \\ldots, Y_{10} \\) be a random sample from \\( N(0,1) \\) (i.i.d). Let, \\[ Y=\\frac{1}{9} \\sum_{i=1}^{9} Y_{i} \\text { and } U=\\sum_{i=1}^{9}\\left(Y_{i}-\\bar{Y}\\right)^{2} \\] What is the distribution of, i) \\( \\bar{Y} \\). ii) \\( U \\). iii) \\( T=\\sum_{i=1}^{9} Y_{i}^{2} \\). iv) \\( V=\\frac{9}{\\sqrt{V+Y_{10}^{2}}} \\). v) \\( W=4 \\frac{F^{2}+Y^{2}}{U} \\). Problem 4. Let \\( Y_{1}, \\ldots, Y_{n} \\) be a random sample with common mean \\( \\mu \\) and variance \\( \\sigma^{2} \\). Use the CLT to write an expression approximating the CDF \\[ P(\\bar{Y} \\leq x) \\] in terms of \\( \\mu, \\sigma^{2}, \\& F_{Z}(z) \\) (the standard normal CDF). 1 of 2 2 Extra Credit Problem 5. [2 points]. Suppose \\( X_{1}, \\ldots, X_{n} \\) and \\( Y_{1}, \\ldots, Y_{m} \\) are two iid random samples from populations with means \\( \\mu_{1}, \\mu_{2} \\) and variances \\( \\sigma_{1}^{2} \\) and \\( \\sigma_{2}^{2} \\), respectively. Show that the distribution function \\( U_{n} \\equiv \\frac{(\\bar{X}-\\bar{T})-\\left(\\mu_{1}-\\rho_{2}\\right)}{\\sqrt{\\left(\\sigma_{1}^{2}+\\sigma_{2}^{2}\\right) / n}} \\) converges to \\( N(0,1) \\) as \\( n \\rightarrow \\infty \\). Problem 6. [3 points]. Let \\( Z \\sim N(0,1) \\) and \\( W \\sim \\chi_{V}^{2} \\) be independent. Find the distribution of \\( T=\\frac{z}{\\sqrt{W / \\nu}} \\).