Question

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

1. The gross weekly sales at a certain super market are a Gaussian random with mean $2200 and standard deviation $230. Assume that the sales from week to week are independent. (a) Find the probability 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 variables each with zero mean and unit variance. Compute that correlation coefficient between

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

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

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

4. Let ? and ? be independent and identically distributed random variables 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 is given by ??(?) = exp{7(?^(?)) − 7} and that of ? by ?? (?) = ( (8/9) (?^(?)) + (6/9))^10 . Assuming that ? and ? are independent, find

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

(b) ?{?? = 0}.

(c) ?(??).

Answer 1

Ans-1

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