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Unformatted text preview: Math 3355 ExamZ Spring 2007 Britt You are to do any 8 of the 9 problems. Clearly denote the omitted problem.
Unsupported work is given very little credit. You may leave factorials in your answer. 1. Consider the random variable X representing the number of 6's obtained when rolling a standard die 3 times.
A) Determine the probability mass function forX.
B) Deﬁne and sketch the distribution ﬁmction for X. 2. The distribution function F (x) for a random variable is given by 0 if x<0 x/3 ifOSx<l
F06): . 2/3 1fl$x<2 l ﬁZSX Compute the following... A) P(X=1) B) P(X<l) C) P(le<2) D) P(X=.25) 3. A coin is biased so that a head is four times as likely to occur as a tail. LetX be the random
variable denoting the number of tails when the coin is ﬂipped twice.
A) Find the expectation and the Variance of X. B) Compute El:X2 — 2X + 6]
C) Compute Var [3X + 71'] 4. Assume X = {0,1, 2, 3, is a random variable with probability mass function given by c— if x=0,l,2,3,... p(x) = x! Determine the value of c and determine the expected value
0 otherwise
of the random variable. 5. A retailer buys electronic devices from a manufacturer. The devices have a random defective
rate of 3%.
A) The retailer inspects 20 items from a shipment. What» is the probability the shipment contains at least one defective item?
B) The retailer gets 10 shipments a month and the inspector randomly tests 20 items from each
shipment. What is the probability there will be three shipments with at least one defective item. More on the Back 6. Suppose a book has 1000 pages and it contains 100 typographical errors. If these errors are
randomly distributed throughout the book, what is the probability that 10 pages, selected at
random, will have no more than one error? Hint: Suppose that X = number of errors per page has a Poisson distribution. 7. Cards are drawn, with replacement, one at a time from a standard deck, until you get a queen.
Let X be the number of draws required. Prove the expected value of X will be 13. You do not
get credit for using the expectation formula from your handout. 8. A die is rolled repeatedly until a 3 appears for the 7th time. Let X be the random variable
denoting the number of rolls required. Setup an expression to calculate P (X Z 25) . 9. A store buys electrical devices in lots of size 10. It is policy to inspect 3 devices randomly
from each order and accept the plot only if all three are nondefective. If 30% of the lots have 4
defectives and 70% of the lots have only one defective, what proportion of the lots does the purchaser reject? BONUS
1. The item produced by a manufacturer has a defective rate of 10%. Find the probability that a
sample of 10 items contains at most one defective using... A) the binomial distribution...
B) approximately, by using the Poissondistribution. 2. Recall that Var(X) = E [(X — ﬂy]. Derive the computational formula Var(X) = E[X2]—(E[X])2 2.. A}?(x=\\ = :PQng—‘Pbub = 2/3—\/3. v3
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 Spring '08
 Britt
 Probability, Probability theory, Probability mass function

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