Probability Distributions
The variables in problem 5.1 are either discrete or continuous. Which are they and why?
Number of siblings -- continuous. This is because the number of siblings cannot change at the moment
Number of conversations -- discrete. There will be more conversations to be held between the children and the mothers.
Explain why the variable "number of dinner guests for Thanksgiving dinner" is discrete.
The variable is discrete because the number of guests can change through cancelation or extra invites.
Explain why the variable "number of miles to your grandmother's house" is continuous.
The variable is continuous because the distance cannot change and the house will not move as it is stationary.
Problem
Express P (x) =1/6; for x = 1, 2, 3, 4, 5, 6, in distribution form.
P (1) = 1/6
P (2) = 2/6 = 1/3
P (3) = 3/6 = 1/2
P (4) = 4/6 = 2/3
P (5) = 5/6
P (6) = 6/6
b. Construct a histogram of the probability distribution P (x) =1/6; for x = 1, 2, 3, 4, 5, 6, in distribution form
1/6
1/6
1/6
4/6
4/6
6/6
c. Describe the shape of the histogram in part (b)
Dog food distribution
Problem 5.18
Find the mean and standard deviation of the number of ships that arrive at a harbor on a given day
x
P (x)
xP (x)
x2
x2 P (x)
10
0.4
4
40
11
0.2
2.2
24.2
12
0.2
2.4
28.8
13
0.1
1.3
16.9
14
0.1
1.2
19.6
60
1
11.1
Mean = ? = ?[xP (x)] = 11.1
Variance = 129.5 -- (11.1)2 = 6.29
Standard Deviation = ?6.29 = 2.508
Problem 5.28
State a very practical reason why the defective item in an industrial situation might be defined to be the "success" in a binomial experiment
Supposing this industrial unit obtained numerous complaints and criticisms. The supervision representatives are coming in to ascertain these criticisms. Taking this into consideration, checking and proving the items and ascertaining them to be defective can be well-defined as 'success' for the government supervision representatives.
Problem 5.48
Consider the binomial distribution where n = 11 and p = 0.05 (see problem 5.47).
a. Use the distribution [problem 5.47(b) or Table 2] and find the mean and standard deviation using formulas (5.1), (5.3a), and (5.4).
Using formula 5.7 and 5.8,
the mean in np = 0.55
and the standard deviation is ?npq
=?11*0.05*0.95 = 0.7228
b. Compare the results of part (a) with the answers found in problem 5.47(a).
The results would have been the same if we had used formulas (5.1), (5.3a), and (5.4).
Probability Concepts & Applications (1) Describe the rationale for utilizing probability concepts. Is there more than one type of probability? If so, describe the different types of probability. One uses probability mathematics in order to assess the probability of a particular occurrence or the results of a particular action; For instance, whether or not one should go into a certain market or invest in a certain product -- what are the chances
An example of a business situation where the probability distribution may be utilized is a scenario where a manager attempts to predict employee retention. The statistical test would be a binary logistic regression analysis. The probability distribution is paramount to this example because it utilizes the dichotomous dependent variable of Yes/No (employee stays/employee quits). Independent variables that may be used to predict employee retention include education level, length of employment,
die is rolled. What is the probability that the number on top is the following? Three can only come about once. Since the chances can be six, the probability is 1/6. An odd number There are three odd numbers, which are 1, 3, and 5. Therefore probability is 1/6 + 1/6 + 1/6 = 3/6 A number less than There are four numbers less than five, which are 1, 2, 3 and 4 Therefore, the
Probability provides a measurable and quantifiable indication of how likely a particular outcome is for an event or experiment. It is expressed as a numeric value between zero and one (inclusive), where an impossible event has a probability of zero and a certain event has a probability of one. Probability is conventionally written using the symbol P, and may be expressed as a percentage by multiplying P. By 100. The problem
(Bienstock, 1996) These are stated to be the reason that these systems "lend themselves to investigation using simulation methodology." (Bienstock, 1996) Simulation is stated to offer an alternative "for understanding these systems, since experimenting with the actual systems would be too costly." (Bienstock, 1996) Furthermore, simulation is stated to facilitate the "examination of dynamic processes or systems over time by allowing the compression of real time." (Bienstock, 1996) Bienstock states
" (Rizzoli, Oliverio, Montemanni and Gambardella, 2004) According to Rizzoli, Oliverio, Montemanni and Gambardella objectives are that which "measure the fitness of a solution. They can be multiple and often they are also conflicting. The most common objective is the minimization of transportation costs as a function of the traveled distance or of the travel time; fixed costs associated with vehicles and drivers can be considered, and therefore the number of
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