Thank you in advance, i'm not so good with this subject so couldyou include the steps you did please

1. The gross weekly sales at a certain super market are aGaussian random with mean $2200 and standard deviation $230. Assumethat the sales from week to week are independent. (a) Find theprobability that the gross sales over the next two weeks exceed$5000. (b) Find the probability that the gross weekly sales exceed$2000 in at least 2 of the next 3 weeks.

2. Let ?1,?2,?3 and ?4 be pairwise uncorrelated random variableseach with zero mean and unit variance. Compute that correlationcoefficient between

(a) ?1 + ?2 and ?2 + ?3. (b) ?1 + ?2 and ?3 + 4.

3. Let the random variables ? be the sum of independent Poissondistributed random variables, i.e., ? = ∑ ? (top) ?=1(bottom) ?? ,where ?? is Poisson distributed with mean ?? .

(a) Find the moment generating function of ?? . (b) Derive themoment generating function of ?. (c) Hence, find the probabilitymass function of ?.

4. Let ? and ? be independent and identically distributed randomvariables with mean ? and variance ?^2. Find the following:

(a) ?[(? + 7) ^2 ]

(b) Var(8? + 9)

(c) ?[(? − ?) ^2 ]

(d) Cov{(? + ?), (? − ?)}

5. The moment generating function of the random variable X isgiven by ??(?) = exp{7(?^(?)) − 7} and that of ? by ?? (?) = ((8/9) (?^(?)) + (6/9))^10 . Assuming that ? and ? are independent,find

(a) ?{? + ? = 7}.

(b) ?{?? = 0}.

(c) ?(??).