Math115a Random Variables #1 Name:________________________Reyes1. Determine if the random variables are finite or continuous:a. X records the length of time in seconds and fractions of seconds between consecutive vehicles that pass by a particular intersection ______________________b. X records the number of hours in a 3 month period in which an individual works at her part time job __________________c. R records the sum of the money from a box of coins __________________2. Let A be the age of a randomly selected student in your English class. You find the ages of seven students sitting in your row are 20, 21, 25, 19, 18, 21, and 20 years.a. Use this information to estimate E(A).b. Find FA(20).c. Give a practical explanation of the value of FA(20) in terms of this specific problem.3. Y is an exponential random variable that records the waiting time in minutes between consecutive customer arrivals ata particular store. The expected value of Y is 12 minutes. Find the following:a. Find the probability that the time between consecutive arrivals of customers is more than 10 minutes.b. Find the probability that the time between consecutive arrivals of customers is exactly 13 minutes.c. 4Yf _____________ d. 4YF _____________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) in this indicate if it is for a discrete or a continuous random variable, or none of these. If the answer is none, explain why.a.00.10.20.30.40.50.61 2 3 4 5 6Series1b.c.x0 1 2 3 4 Xf x0.1 0.1 0.3 0.4 0.1d.e. f.5. Let X be a random variable that records the number of heads in 4 tosses of a coin.a. What is the sample space for this experiment?b. Is this a discrete random variable or a continuous random variable? Explain.c. Is this an example of a Binomial Random Variable? Explain.d. Use a graph to illustrate and ( )X Xf x F x.e. 3 =Xf_____________ d. (3)XF _____________6. A supplier of kerosence has a 150 gallon tank that is filled at the beginning of each week. His weekly demand shows a relative frequency behavior that increases steadily up to 100 gallons and then levels off between 100 and 150 gallons. If Y denotes weekly demand in hundreds of gallons, the pdf is given by: 0 11 1 1.50 elsewhereYy yf y y a. Verify that Yf y is a pdf.b. Determine the probability that the weekly demand is greater than 1.c. Determine the probability that the weekly demand is less than 1.7. 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. 2P X b. 1 2P X c. 1 4P X d. 4P X 8. A baseball player is a 0.300 hitter. He comes to bat three times in a game.a. How many hits do we expect the player to get?b. What is the probability that he will get exactly one hit?9. Suppose that X is exponentially distributed with a mean 1/2. What is P(X>1)?10. You were told that a car broke down between 2:00 pm and 5:00 pm and that any time within this interval is likely to be the exact failure time of the car. Thus you would use the following probability density function, 132 50 otherwiseXxf x where x is the exact time the car failed between 2:00 and 5:00 pm.a. What is the probability that a car failed after 3:00?b. What is the probability that a car failed at exactly 2:30?c. What is the probability that a car failed between 2:20 and 4:00?11. The probability that a patient recovers from a delicate heart operation is 0.9. There are 4 patients in ICU recovering from their heart surgery. Let X record the number of patients that survive. a. Give the probability mass function for X.b. Give the cumulative density function for X.c. What is the probability that at most two survive?d. 4Xf ____________ e. 3XF
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