I\'m reviewing for my first midterm and these are two old HW problems that I was
ID: 3124791 • Letter: I
Question
I'm reviewing for my first midterm and these are two old HW problems that I wasn't able to solve. Please show your work, and be as clear as possible. Thanks!
2. A nefarious gambler has developed a weighted six-sided die, with sides marked 1,2,3,4,5, and 6. The proba- bility that a 6 is rolled is 0.5, and all the other sides are equally likely. Define the random variable X = the value on the die when it is rolled once. (a) Write out the pmf of X. (b) Compute the probability that an even number is rolled. c) Compute the expectation and variance of XExplanation / Answer
2.
a)
Let p = the probability of each of the other sides
As all probabilities must add up to 1, then
p + p + p + p + p + 0.5 = 1
5p + 0.5 = 1
p = 0.1
Hence,
p(x) = 0.1, x = 1,2,3,4,5
0.5, x = 6
0, otherwise [ANSWER]
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b)
P(even) = P(2) + P(4) + P(6) = 0.1+0.1+0.5 = 0.7 [ANSWER]
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c)
Consider:
Thus,
E(x) = Expected value = mean = 4.5 [ANSWER, EXPECTATION]
Var(x) = E(x^2) - E(x)^2 = 3.25 [ANSWER, VARIANCE]
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Hi! Please submit the next part as a separate question. That way we can continue helping you! Please indicate which parts are not yet solved when you submit. Thanks!
x P(x) x P(x) x^2 P(x) 1 0.1 0.1 0.1 2 0.1 0.2 0.4 3 0.1 0.3 0.9 4 0.1 0.4 1.6 5 0.1 0.5 2.5 6 0.5 3 18