1. Let X denote the number of rainy days on the coming weekend. Suppose that the
ID: 1229480 • Letter: 1
Question
1. Let X denote the number of rainy days on the coming weekend. Suppose that there is a 20% chance of rain on Saturday and 50% chance of rain on Sunday. Assume that rain on Saturday and rain on Sunday are independent events, so that Prob(rain on Saturday AND rain on Sunday) = Prob(rain on Saturday)Prob(rain on Sunday), Prob(no rain on Saturday AND rain on Sunday) = Prob(no rain on Saturday)Prob(rain on Sunday), and so on.a. Explain why X is a random variable.
b. Write the PDF of X.
c. Let F(x) denote the CDF of X evaluated at X = x. Find F(1).
d. Find E(X).
e. Find Var(X).
Explanation / Answer
Probability it will not rain on Saturday: 1-.5 = .5 or 50%
Probability it will not rain on Sunday: 1-.5 = .5 or 50%
Probability it will not rain on Saturday or Sunday: P(not rain on Sat)*P(not rain on Sun)
= .5*.5 = .25 or 25%
Probability it will rain on the weekend = 1-P(not rain on Sat or Sun) = 1-.25 = .75 or 75%
To check:
Probability it rains on Saturday but not Sunday: P(rain on Sat)*P(not rain on Sun)
= .5*.5 = .25 or 25%
Probability it doesn't rain on Saturday but does on Sunday: P(not rain on Sat)*P(rain on Sun)
= .5*.5 = .25 or 25%
Probability it rains on Saturday and Sunday: P(rain on Sat)*P(rain on Sun)
= .5*.5 = .25 or 25%
P(rains on weekend) = P(Sat but not Sun) + P(Sun but not Sat) + P(Sat and Sun)
= .25 + .25 + .25 = .75
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