What are the probability distribution of X and Y? Are they independent?

Data were collected from a survey given to graduating college seniors on the number of times they had changed majors. From that data, a probability distribution was constructed. The random variable X is defined as the number of times a graduating senior changed majors. It is shown below: 1 3 4 8 P(X = x) |0.252 0.286 0.13 0.188 0.082 0.038 0.02 0.003 0.001 a. What is the probability that a randomly selected student changed his or her major at least once? b. What is the probability that a randomly selected student changed his or her major at most twice? c. Given that a randomly selected person did change majors, what is the probability that he or she changed majors more than three times? (Round your answer to three decimal places.)

The table below shows the grade distribution for a section in 6 classes. Determine if each of the distributions 1-6 are valid probability distributions or not and explain your reasoning. Distribution A B C D E 1 0.3 0.3 0.3 0.2 0.1 2 0 0 1 0 0 3 0.3 0.3 0.3 0 0 4 0.3 0.5 0.2 0.1 -0.1 5 0.2 0.4 0.2 0.1 0.1 6 0 -0.1 1.1 0 0

Suppose that the random variable x, shown below, represents the number times. P(x) represents the probability of a randomly selected person having received that number of speeding tickets during that period. Use the probability distribution table shown below to answer the following questions. 43 x P(x) = 0 1 2 3 > Next Question 4 5 6+ 0.2951 0.2587 0.1924 0.1604 a) What is the probability that a randomly selected person has received five tickets in a three-year period? P(x = 5) 0.0492 0.0442 0.0000 b) What is the probability that a randomly selected person has received one tickets in a three-year period? P(x = 1) = c) What is the probability that that a randomly selected person has received more than zero tickets in a three- year period? P(x > 0) d) What is the probability that that a randomly selected person has received one or less tickets in a three-year period? P(x ≤ 1) =