Download Homework due 09/15 1. Consider a sequence of five Bernoulli trials

Homework due 09/15
1. Consider a sequence of five Bernoulli trials. Let X be the number of times that a head is
followed immediately by a tail. For example, if the outcome is ω = HHT HT then X(ω) = 2
since a head is followed directly by a tail at trials 2 and 3, and also at trials 4 and 5. Find
the probability mass function of X, assume that p = 12 .
2. We roll a fair die three times. Let X be the number of times that we roll a 6. What is
the probability mass function of X?
3. Some day, 10,000 cars are travelling across a city. Suppose that the probability that a car
has an accident this day is 0.002. Using the approximation of a binomial distribution by a
Poisson distribution, compute the probability that exactly 15 cars have an accident this day.
4. Let F be the function defined by:
F (x) =

0


2


 x3
1
3

1


x+

 6
1
1
3
x<0
0≤x<1
1≤x<2
2≤x<4
4≤x
Let X be a random variable which corresponds to F .
1. Verify that F is a cumulative distribution function. (You need to verify the conditions
(a)-(c) we mentioned in class.)
2. Compute P (X = 2).
3. Compute P (X < 2).
4. Compute P (X = 2 or
1
2
≤ X < 32 ).
5. Compute P (X = 2 or
1
2
≤ X ≤ 3).
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