143 , Measures of D perion Problem 11: A fair six-sided die is tossed. You win $
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143 , Measures of D perion Problem 11: A fair six-sided die is tossed. You win $2 if the result is a 1, you win $1 if the result is a 6. Otherwise, you will lose $1. The probability distribution for x-am won or lost is Probability 0316 1/6 1/6 a. Verify that X follows a discrete distribution. (Q points) b. Calculate the expected value of X. Provide a practical interpretation of the result (4 points: 2 points for calculations and 2 points for interpretation) . Ib7 is "expected Value Problem 12: (4 points each) Given a set of data points: A single number is drawn from this data set. 1 3 4 4 79 10 12 13 14 a. What is the probability the number is 4 or odd? (4 points) (4 points) b. What is the probability the number is 7 or 9? c. What is the probability the number is 7 and odd? (4 points)Explanation / Answer
11. (a) A random variable X is said to be discrete if it can take on only a finite number—or a countably infinite number—of possible values x.
We can see that X can take values =2, =1 or -1 and no other values. Hence X is a discrete variable.
The probability mass function satisfies the two required criteria:
(i) P(X = x) = p(x) 0
(ii) P (X = x) = 1 , where the sum is over all possible values of x.
The probability mass function P (X=x) can take only 2 values 1/6 and 4/6 and nothing else.
Hence we can conclude that X follows a discrete distribution.
(b) E(X) = 2 x (1/6) + 1 x (1/6) - 1 x (4/6) = (2+1-4) / 6 = - 1/6 = =0.167
The interpretation of this is that: Over a large number of rolls of the dice, the average loss is $ 0.167.
12. (a) P (X=4 or odd) = P(X=4) + P(X is odd)
P(X=4) = 2/10, since there are 2 occurences of the number 4, so 4 is twice as likely to occur as the other numbers.
P(X is odd) = P(X=1,3,7,9,13) = 5/10
P (X=4 or odd) = P(X=4) + P(X is odd) = 2/10 + 5/10 = 7/10 = 0.7
(b) P(X=7 or 9) = P(X=7) + P(X=9) = 1/10 + 1/10 = 0.2
(c) P(X=7 and X is odd) : There is only 1 value X=7 which satisfies the criteria of both being odd and having a value of 7
P(X=7 and X is odd) = 1/10
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