1-Verify that the following function is a probability mass function, and determi
ID: 3125092 • Letter: 1
Question
1-Verify that the following function is a probability mass function, and determine the respective probabilities
x
-2
-1
0
1
2
f(x)
1/8
2/8
2/8
2/8
1/8
P(X2)
P(X> -2)
P (-1 X 1)
P(X -1 or X = 2 )
2-Determine the cumulative distribution function for the random variable in (Q1) above and determine the following probabilities
a. P(X1.25)
b. P(X2.2)
c. P (-1.1 < X 1)
d. P (X > 0)
3-Heart failure is due to either natural occurrences (87%) or outside factors (13%). Outside factors are related to induced substances or foreign objects. Natural occurrences are caused by arterial blockage, disease, and infection. Suppose that 20 patients will visit an emergency room with heart failure. Assuming that causes of heart failure between individuals are independent;
What is the probability that three have conditions caused by outside factors?
What is the probability that three or more individuals have conditions caused by outside factors?
What are the mean and standard deviation of the number of individuals with conditions caused by outside factors?
4-The number of telephone calls that arrive at a phone exchange is often modeled as a Poisson random variable. Assuming that on average there are 10 calls per hour;
What is the probability that there are exactly five calls in one hour?
What is the probability that there are three or fewer calls in one hour?
What is the probability that there are exactly fifteen calls in two hours?
5-The number of cracks in a section of interstate highway that are significant enough to require repair is assumed to follow a Poisson distribution with a mean of two cracks per mile.
What is the probability that there are no cracks that require repair in 5 miles of highway?
What is the probability that at least one cracks requires repair in 1/2 mile of highway?
If the number of cracks is related to the vehicle load on the highway and some sections of the highway have a heavy load of vehicles whereas other sections carry a light load, how do you feel about the assumption of a Poisson distribution for the number of cracks that require repair?
x
-2
-1
0
1
2
f(x)
1/8
2/8
2/8
2/8
1/8
Explanation / Answer
1-Verify that the following function is a probability mass function, and determine the respective probabilities
x
-2
-1
0
1
2
f(x)
1/8
2/8
2/8
2/8
1/8
Since sum of the probabilites is
(1/8) + (2/8) + (2/8) + (2/8) + (1/8) = 1
So it is pmf.
Now
P(X2) = P(X=-2) + P(X=-1) + P(X=0) + P(X=1) + P(X=2) = 1
P(X> -2) = P(X=-1) + P(X=0) + P(X=1) + P(X=2) = 7/8
P (-1 X 1) = P(X=-1) + P(X=0) + P(X=1) = 6/8
P(X -1 or X = 2 ) = P(X=-2) + P(X=-1) + P(X=2) = (1/8) + (2/8) + (1/8) = 4/8
2.
CDF will be
F(-2) = P(X -2) = 1/8
F(-1) = P(X -1) = P(X=-2) + P(X=-1) = 3/8
F(0) = P(X 0) = P(X -1)+P(X=0) = 5/8
F(1) = P(X 1) = P(X 0)+P(X=1) = 7/8
F(2) = P(X 2) = P(X 1)+P(X=2) = 1
a. P(X1.25) = F(1) = 7/8
b. P(X2.2) = F(2) = 1
c. P (-1.1 < X 1) = F(1) - F(-1) = 4/8
d. P (X > 0) = 1- F(0) = 4/8
x
-2
-1
0
1
2
f(x)
1/8
2/8
2/8
2/8
1/8
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