Statistics & Probability

1 ) Suppose the parameter T can take the values in the table below with the corresponding probabilities:

VALUES PROBABIITTY 10 0.45 20 0.05 30 0.18 40 0.01 1,000,000 0.01 Calculate the mean, mode, and median of the distribution of T 
 Which single value would you use to summarize the distribution if you had to? What would be the advantages and disadvantages of doing so?

2 ) Suppose a particle beam is being fired at a target consisting of several layers of material. Let F 
 represent the percentage of particles that will pass through any given layer if given the chance. At the beginning of the experiment, you have no knowledge of F 
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and believe it could be any value from 0 to 1 uniformly. Using Laplace's Rule of Succession, calculate the probability that a given particle will pass through 3 layers of material.

3 ) One of the original examples Pierre-Simon Laplace gave of his Rule of Succession was the "sunrise problem": Letting p 
 be the chance of the sun rising each day,

after observing it having risen 10,000 days in a row, using the rule of succession we would say the probability of it rising tomorrow is 10001 
/10002 

≈0.99990002
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 Laplace was roundly criticized for this, because it seems ridiculous to say that the sun rising is anything less than certain. What did his critics misunderstand about the assumptions Laplace was using to do this calculation?