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In the set of the first 20 natural numbers, i,e, {1,2,3,…,20}, four distinct numbers are to be drawn randomly. Among the four numbers drawn, denote by X the largest number, and by Y the second largest number.

(a) Write down the value of Pr(X=3),
(b) Write down the value of Pr(Y=20).
(c) Find the probability mass function of X. State the domain clearly.
(d) Find the probability mass function of Y. State the domain clearly.
(e) Find the cumulative distribution function of X.
(f) Find the cumulative distribution function of Y.

User Tpeczek
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Final answer:

The probability of X being equal to 3 is 0.05 and the probability of Y being equal to 20 is also 0.05. The probability mass function of X and Y is uniform, with a probability of 1/20 for each value. The cumulative distribution function of X and Y can be represented as step functions.

Step-by-step explanation:

a. The probability of X being equal to 3 can be calculated by dividing the number of favorable outcomes (in this case, the number 3) by the total number of possible outcomes (20). So, Pr(X=3) = 1/20 = 0.05.

b. The probability of Y being equal to 20 can be calculated by dividing the number of favorable outcomes (in this case, the number 19) by the total number of possible outcomes (20). So, Pr(Y=20) = 1/20 = 0.05.

c. The probability mass function of X is a function that assigns probabilities to each possible value of X. In this case, X can take on any value from 1 to 20. Since every value has an equal probability of being drawn, the probability mass function is uniform, with a probability of 1/20 for each value of X.

d. The probability mass function of Y is also uniform, with a probability of 1/20 for each value of Y from 1 to 19.

e. The cumulative distribution function of X gives the probability that X is less than or equal to a certain value. In this case, since the probability of each value of X is 1/20, the cumulative distribution function of X can be represented as a step function, with a step of 1/20 at each value of X from 1 to 20.

f. The cumulative distribution function of Y can be represented as a step function as well. It starts at 0 for Y=1 and increases by 1/20 at each value of Y from 1 to 19.

User Mask
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