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Practice for Midterm 1 1. Let X and Y be independent random variables with variances V ar(X) = 4 and V ar(Y ) = 3. What is the standard deviation of the difference, X − Y ? 2. Let X be a random variable with probability mass function pX given as k pX (k) -1 0.2 0 0.6 1 0.2 What is the variance of X? 3. A manufacturing process produces integrated circuit chips. The fraction of bad chips produced is 20%. Thoroughly testing a chip to determine whether it is good or bad is expensive, so a cheap test is tried. All good chips will pass the cheap test, but so will 10% of the bad chips. (a) Given a chip passes the cheap test, what is the probability that it is a good chip? (b) If a company using this manufacturing process sells all chips which pass the cheap test, what percentage of chips sold will be bad? 5. How many anagrams (= different “words”) can be built from the letters ‘HOT CHOCOLATE’ ? Do not ignore the space and count rearrangements of these letters so that there are two distinct words (each must have at least one letter). 6. Some jobs submitted for processing on a particular CPU have fatal programming errors, while others do not. Suppose that the long run fraction of jobs with fatal programming errors is p = 0.05. (a) Find the probability that among the next 10 jobs submitted there are less than 3 with fatal errors. 1. Discrete Distributions Given is the following table: x pX −3 0.1 −1 0.15 0 0.3 1 s (a) Find s, so that pX is a probability mass function. (b) Assume, the random variable X has probability mass function pX . What is E[X]? (c) Determine V ar[X]. (d) Assume, Y is another random variable with E[Y ] = 1.25 and V ar[Y ] = 2.5. Which of the following expressions can be derived? Say yes or no, if possible, give the result: i. E[4X + Y ] ii. V ar[X + 0.5Y ] iii. V ar[3Y ]