STAT 441: Homework 1 Due: Monday, 02/06/2023 by 11:59 pm STAT 441: Homework 1 Due: Monday, 02/06/2023 by 11:59 pm 1. Let ���!, ���”, and ���# be independent standard normal random variables. Let ��� = ���” + ���” + ���” !”# a) Find ���(��� > 3) (You may use R to find the probability) b) Find ���(���) and ���ar(���). 2. Let ���! and ���” be independent random variables. The distribution of ���! is normal with mean ��� = 1 and variance ���” = 2. The distribution of ��� is also normal with mean ��� = 2 and !!”” variance ���” = 4. ” a) Let ��� = ���! + ���”. Find ���$(���), the moment-generating function (MGF) of ���. b) Based on the MGF found in part a), what is the probability distribution of ���? 3. Let ���! and ���” be independent random variables. Let ��� = ���! + ���”. Suppose ��� has a chi- square distribution with ��� degrees of freedom, ���! has a chi-square distribution with ���! degrees of freedom, and ���! < ���. Show that ���” has a chi-square distribution with (��� − ���!) degrees of freedom. 4. Suppose the amount dispensed by a filling machine is regulated to have a normal distribution with mean ��� ounces per bottle and standard deviation ��� = 1 ounce. The mean ��� is unknown. A sample of ��� = 9 filled bottles is randomly selected from the output of the machine and the ounces of fill are measured for each. Then the mean of the sample is used to estimate the value of ���. Find the probability that the sample estimate will be within 0.3 ounces of the true value of ���. 5. Let ���!,���”,…,���#& be a random sample of size ��� = 30 from the Poisson distribution with parameter ��� = 1. a) Let ��� = ∑#& ��� . What is the probability distribution of ���? Provide the distribution’s ‘(! ‘ name and the parameter value. b) Let ���B = $ = ! ∑#& ��� be the sample mean. Use the result in part a) to find the exact #&B#& ‘(! ‘ value of ���(��� ≤ 1.2). Round the answer to 2 decimals. c) Use the CLT to find an approximate value of ���(���B ≤ 1.2). Round the answer to 2 decimals. Is the approximation close? You may use R to calculate probability with the respect to the Poisson distribution, In R, ppois(q, lambda) will provide p(X≤ q) for X~ Poisson(lambda). STAT 441: Homework 1 Due: Monday, 02/06/2023 by 11:59 pm 6. Suppose the length of a leaf of a certain full-grown plant has a normal distribution with mean ��� = 4 inches and variance ���” = 0.25. What is the probability that the variance in length for a random sample of 20 leaves is a) less than 0.5? b) greater than 0.25? (You may use R to calculate the probabilities.)