Statistics?

1) Let a card be selected from an ordinary deck of playing cards. The outcome c is one of these 52 cards. Let X(c) = 4 if c is an ace, let X(c) = 3 if c is a king, let X(c) =2 if c is a queen, let X(c) =1 if c is a jack, and let X(c) = 0 otherwise. Suppose that P assigns a probability of 1/52 to each outcome c. Describe the induced probability P_x(D) on the space D= {0,1,2,3,4} of the random variable X.


2) Consider an urn which contains slips of paper each with one of the number 1,2,...,100 on it. Suppose there i slips with the number i on it for 1 = 1,2,...1,00. For example, there are 25 slips of paper with the number 25. Assume that the slips are identical except for the numbers. Suppose on slip is drawn at randọm Let X be the number on the slip.


(a) Show that X has the pmf p(x) = x/5050, x = 1,2,3,...,100, zero elsewherẹ


(b) Show that the cdf of X is F(x) = [x]([x] +1)/10100, for 1%26lt;/= x =%26lt;100, where [x] is the greatest integer in x

Statistics?
PX(1) = 4/52 ; PX(2) = 4/52 ; PX(3) = 4/52 ; PX(4) = 4/52


[4/52 = 1/13] ; PX(0) = 36/52 or 9/13





If there are I slips for the I-th number, the total number of slips in the urn is 1+2+3+...+99+100 = 5050. So, the chance of drawing I is I/5050.





What is the sum of the numbers 1 through N?


N(N+1)/2





This should give you a clue to the second part of the question.