Math115a Probability Distribution #2 Name:________________________1. Determine if the following are random variables, if they are indicate whether they are finite or continuous:a. X records the outcomes of a soccer game, the outcomes are “win”, “lose” or “tie” ______________________b. X records the exact amount of time it takes to get from point A to point B __________________c. R records the distance between the location where a ball lands and the location from where the thrower was standing __________________d. W records the sum of the card faces in a set of five playing cards __________________e. X records the exact time of message arrivals at a communications network node during a one minute period___________2. A coin is tosses twice. Let X denote the number of heads. a. What are the possible values for X?b. Is X a discrete random variable? Explain.c. Is X a uniformly distributed random variable? Explain.3. A basketball player throws four free throws in a game. His free throw shooting percentage is 0.75. a. How many free throws do we expect him to hit? b. Show a table of Xf x and XF x.c. What is the probability that he will hit exactly two shots?d. What is the probability that he will hit at most two shots?4. For each of the following, decide which could be a p.m.f. (Probability mass function), a p.d.f. (Probability density function), a c.d.f. (Cumulative density function), or none of these. If the answer is none, explain why.a. 0 00.2 0 50 5xp x xx b.c. d.e. f. 0 00 331 3xxp x xx 5. Suppose that X is exponentially distributed with a mean 1. What is P(X>1)?6. X is a random variable with the following pdf: 1212322 01 12 3 10 elsewhereXx xxf xx x Verify that Yf y is a pdf.7. Suppose X is a continuous random variable and the density curve is given in graphical form. Shade in the region under the following density curves that correspond to the following probabilities:a. 8P X b. 3 8P X c. 2P X d.(2 3.2)P X 8. Suppose that X is an exponential random variable, which has the following pdf and cdfs. 60 0106xXxf xe x 60 01 0xXxF xe x a. Determine the mean.b. Find and interpret: 4XFc.(1 5)P x d. Find and interpret: 4Xfe. Is 4 ( 4)Xf P X ? Explain.9. Suppose that X is a uniform random variable with an expected value of 10. a. Determine equations for Xf x and XF x.b. 6XFc. 6Xf10. Let X be a binomial random variable with n=5 and p=0.4. Find E(X).11. Let X be a binomial random variable with n=100. If E(X)=3, determine
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