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Homework answers / question archive / 1) All about the random variables 1

1) All about the random variables 1

Statistics

1) All about the random variables

1.1

Suppose that on the average a certain store serves 15 customers per hour. What is the probability that the store will serve more than 20 customers in a particular two-hour period? [HINT: Customer per hour is a rate. You might want to look for distributions dealing with that type of data.]

1.2

Suppose that a certain examination is to be taken by five students independently of one another, and the number of minutes required by any particular student to complete the examination has an exponential distribution for which the mean is 80. Suppose that the examination begins at 9:00 AM. Determine the probability that at least one of the students will complete the examination before 9:40 AM.

1.3

Sketch the pdf of the Gamma distribution for the following pairs of values of the parameters a = ½ and β = 1. [HINT: Check what parameters a and β refer to, and check the documentation in R for rgamma and dgamma]

1.4

Suppose that the measured voltage in a certain electric circuit has a normal distribution with mean 120 and standard deviation 2. If three independent measurements of the voltage are made, what is the probability that all three measurements will lie between 116 and 118°?

2 R Exercise

You have a bag with 5 cats in it. 3 of the cats in the bag are black, and 2 of the cats are white. When you open the bag, one of the cats darts out and climbs a tree before you can see what kind of cat it is (black/white). Say you pull another cat out of the bag and see that it is a white cat.

What is the probability that the cat in the tree is white? After solving this manually, write a function in R simulating the situation 5000 times and returns (roughly) the probability that you calculated. [HINT: This is all about problem solving. Think carefully about what steps you need to answer the question and “translate” them into instructions that R can understand.]

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