10/6/23, 9:35 AM
Quiz: Activity 1b (Discrete Probability Distribution)
Activity 1b (Discrete Probability Distribution)
Started: Oct 6 at 9:24am
Quiz Instructions
Answer the following. Show your complete and clear solution.
Question 1
5 pts
Actual lengths of stay at a hospital’s emergency department in 2009 are shown in the following table (rounded to the nearest hour).
Length of stay is the total of wait and service times. Some longer stays are also approximated as 15 hours in this table.
Hours
Count
Percent
1
19
3.80
2
51
10.20
3
86
17.20
4
102
20.40
5
87
17.40
6
62
12.40
7
40
8.00
8
18
3.60
9
14
2.80
10
11
2.20
15
10
2.00
Calculate the probability mass function of the wait time for service.
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n = 500 // Total Count
Let X denote the wait and service times
PMF:
P(X=1) = 19/500
P(X=2) = 51/500
P(X=3) = 86/500
P(X=4) = 102/500
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10/6/23, 9:35 AM
Quiz: Activity 1b (Discrete Probability Distribution)
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Question 2
6 pts
In a semiconductor manufacturing process, three wafers from a lot are tested. Each wafer is classified as pass or fail. Assume that the
probability that a wafer passes the test is 0.8 and that wafers are independent.
(a) What is the formula for the probability mass function?
3Cx (0.8)^x (0.2)^(3
(b) Determine the probability mass function value (up to 3 decimal place) of the number of wafers from a lot that pass the test.
X=x
P (X = x)
0
0.008
1
0.096
2
0.384
3
0.512
(c) The cumulative distribution function of the number of wafers from a lot that pass the test are shown below. Determine the values of a
to e (up to 3 decimal place
a= 0
b = 0.008
c = 0.104
d = 0.488
e= 1
Question 3
10 pts
The thickness of wood paneling (in inches) that a customer orders is a random variable with the following cumulative distribution function:
(a) Determine P(X ≤ 1/18) = 0
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10/6/23, 9:35 AM
Quiz: Activity 1b (Discrete Probability Distribution)
(b) Determine P(X ≤ 1/ 4) = 0.9
(c) Determine P(X ≤ 5 /16) = 0.9
(d) Determine P(X > 1/ 4) = 0.1
(e) Determine P(X ≤ 1/ 2) = = 1
Question 4
9 pts
Determine the mean, variance and standard deviation of the random variable with the given probability mass function by completing the
table below ( answer up to 2 decimal place).
X=x
P (X=x)
0
0.04
0.00
0.00
1
0.12
0.12
0.12
2
0.20
0.40
0.80
3
0.28
0.84
2.52
4
0.36
1.44
5.76
2.8
9.2
TOTAL
2.80
1.36
1.17
Question 5
6 pts
In a NiCd battery, a fully charged cell is composed of nickelic hydroxide. Nickel is an element that has multiple oxidation states. Assume
the following proportions of the states:
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10/6/23, 9:35 AM
Quiz: Activity 1b (Discrete Probability Distribution)
(a) Determine the cumulative distribution function of nickel charge. (values up to 2 decimal palce)
a = 0.00
b = 0.17
c = 0.52
d = 0.85
e= 1
(b) Determine the mean, variance and standard deviation of the nickel charge.
2.29
(up to 2 decimal place)
1.5259
1.2353
(up to 4 decimal place)
(up to 24decimal place)
Question 6
8 pts
The random variable X has a binomial distribution with n = 10 and p = 0.5. Determine the following probabilities (up to 4 decimal place):
(a) P(X = 5) = 0.2461
(b) P(X ≤ 2) = 0.9893
(c) P(X ≥ 9) = 0.0107
(d) P(3 ≤ X < 5) = 0.3223
Question 7
3 pts
Determine the cumulative distribution function of a binomial random variable with n = 3 and p =0.25. (answer up to 6 decimal place)
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10/6/23, 9:35 AM
Quiz: Activity 1b (Discrete Probability Distribution)
a = 0.000000
b = 0.421875
c = 0.843750
d = 0.984375
e = 1.000000
Question 8
9 pts
The phone lines to an airline reservation system are occupied 40% of the time. Assume that the events that the lines are occupied on
successive calls are independent. Assume that 10 calls are placed to the airline.
(a) What is the probability (up to 4 decimal place) that for exactly three calls, the lines are occupied?
0.2150
(b) What is the probability (up to 4 decimal place) that for at least one call, the lines are not occupied?
0.9940
(c) What is the expected number of calls in which the lines are all occupied?
4
Question 9
8 pts
Suppose that X has a Poisson distribution with a mean of 4. Determine the following probabilities (up to 6 decimal place):
(a) P(X = 0) = 0.018316
(b) P(X ≤ 2) = 0.238103
(c) P(X = 4) = 0.195367
(d) P(X = 8) = 0.029770
Question 10
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4 pts
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10/6/23, 9:35 AM
Quiz: Activity 1b (Discrete Probability Distribution)
Suppose that the number of customers who enter a bank in an hour is a Poisson random variable, and suppose that P(X = 0) = 0.05.
Determine
a. the mean of X = 2.995732
(up to 6 decimal place)
b. variance of X = 2.995732
(up to 6 decimal place)
Question 11
4 pts
When a computer disk manufacturer tests a disk, it writes to the disk and then tests it using a certifier. The certifier counts the number of
missing pulses or errors. The number of errors on a test area on a disk has a Poisson distribution with λ = 0.2.
(a) What is the expected number of errors per test area?
0.2
(b) What percentage (up to decimal place) of test areas have two or fewer errors?
99.88
%
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