Complete the following matrix/table, insert into discussion. Find one journal ar
ID: 2766531 • Letter: C
Question
Complete the following matrix/table, insert into discussion. Find one journal article or news article from the NAU online library that uses or discusses one of the distributions.
a) attach a link to the article, or upload the article to the discussion board
b) provide a summary of the article in one paragraph.
In your response to this discussion, read one article from your classmates and comment on the use of the probability distribution, and the article in general.
Type of Distribution
3 key characteristics
Specific Uses
Key equation
Binomial
1) characterized by Bernoulli process which has only 2 possible outcomes, a success and a failure
2)Probability stays the same - it is static
3) Trials are statistically independent - they do not influence each other
4) Number of trials is positive - it cannot be a negative number
Use in manufacturing to explore probability of defect in an item that is made on an assembly line. A random sample would be drawn once an hour and tested for defect. What is the probability of finding a defect Note: a binomial p distribution table must be used. Sample size is on far left signified by n (number of trials), and r = defects found, p = probability of finding the defect.
n = # trials
p = probability of a failure
Normal
F
Exponential
Poisson
Type of Distribution
3 key characteristics
Specific Uses
Key equation
Binomial
1) characterized by Bernoulli process which has only 2 possible outcomes, a success and a failure
2)Probability stays the same - it is static
3) Trials are statistically independent - they do not influence each other
4) Number of trials is positive - it cannot be a negative number
Use in manufacturing to explore probability of defect in an item that is made on an assembly line. A random sample would be drawn once an hour and tested for defect. What is the probability of finding a defect Note: a binomial p distribution table must be used. Sample size is on far left signified by n (number of trials), and r = defects found, p = probability of finding the defect.
n = # trials
p = probability of a failure
Normal
F
Exponential
Poisson
Explanation / Answer
Answer
Type of Distribution
3 key characteristics
Specific Uses
Key equation*
Binomial
1) characterized by Bernoulli process which has only 2 possible outcomes, a success and a failure
2)Probability stays the same - it is static
3) Trials are statistically independent - they do not influence each other
4) Number of trials is positive - it cannot be a negative number
Use in manufacturing to explore probability of defect in an item that is made on an assembly line. A random sample would be drawn once an hour and tested for defect. What is the probability of finding a defect Note: a binomial pdistribution table must be used. Sample size is on far left signified byn (number of trials), and r =defects found,p = probability of finding the defect.
n = # trials
p = probability of a failure
Normal
1) All normal curves are bell-shaped with points of inflection at ± .
2) All normal curves are symmetric about the mean .
3) The area under an entire normal curve is 1.
4) All normal curves are positive for all x. That is, f(x) > 0 for all x.
5) The height of any normal curve is maximized at x = µ.
6) The shape of any normal curve depends on its mean and standard deviation .
Modern portfolio theory assumes returns of diversified asset portfolio follow a normal distribution
In operations management, process variations often are normally distributed.
µ is the mean and 2 is the variance of the distribution. The notation N(µ, 2 ) should be mentioned.
F
(1) F-distributions are generally skewed.
(2) The probability density function of an F random variable with r1 numerator degrees of freedom and r2 denominator degrees of freedom is:
f(w)=(r1/r2)r1/2[(r1+r2)/2]w(r1/2)1[r1/2][r2/2][1+(r1w/r2)](r1+r2)/2f(w)=(r1/r2)r1/2[(r1+r2)/2]w(r1/2)1[r1/2][r2/2][1+(r1w/r2)](r1+r2)/2
over the support w 0.
(3) The definition of an F-random variable:
F=U/r1V/r2F=U/r1V/r2
implies that if the distribution of W is F(r1, r2), then the distribution of 1/W is F(r2, r1).
The F-distribution arises from inferential statistics concerning population variances. More specifically, we use an F-distribution when we are studying the ratio of the variances of two normally distributed populations.
The F-distribution is not solely used to construct confidence intervals and test hypotheses about population variances. This type of distribution is also used in one factor analysis of variance (ANOVA). ANOVA is concerned with comparing the variation between several groups and variation within each group. To accomplish this we utilize a ratio of variances. This ratio of variances has the F-distribution. A somewhat complicated formula allows us to calculate an F-statistic as a test statistic.
F-distribution with r1 numerator degrees of freedom and r2 denominator degrees of freedom. We write F ~ F(r1,r2).
Exponential
Suppose that XX takes values in [0,)[0,). Then XX has the memoryless property if the conditional distribution of XsXs given X>sX>s is the same as the distribution of XX for every s[0,)s[0,). Equivalently,
P(X>t+sX>s)=P(X>t),s,t[0,)
XX has a continuous distribution and there exists r(0,)r(0,) such that the distribution function FF of XX is
F(t)=1ert,t[0,)
In the gamma experiment, set n=1n=1 so that the simulated random variable has an exponential distribution. Vary rr with the scroll bar and watch how the shape of the probability density function changes. For selected values of rr, run the experiment 1000 times and compare the empirical density function to the probability density function.
In the special distribution calculator, select the exponential distribution. Vary the scale parameter (which is 1/r1/r) and note the shape of the distribution/quantile function. For selected values of the parameter, compute a few values of the distribution function and the quantile function.
The exponential distribution can be used as a good approximate model for the time until the next phone call arrives.
Reliability theory and reliability engineering also make extensive use of the exponential distribution.
In hydrology, the exponential distribution is used to analyse extreme values of such variables as monthly and annual maximum values of daily rainfall and river discharge volumes.
Poisson
Estimating the number of car crashes in a city of a given size; in physiology,
Calculation of the probabilistic frequencies of different types of neurotransmitter secretions.
*Key equation format cannot be supported and presented in proper manner.
Type of Distribution
3 key characteristics
Specific Uses
Key equation*
Binomial
1) characterized by Bernoulli process which has only 2 possible outcomes, a success and a failure
2)Probability stays the same - it is static
3) Trials are statistically independent - they do not influence each other
4) Number of trials is positive - it cannot be a negative number
Use in manufacturing to explore probability of defect in an item that is made on an assembly line. A random sample would be drawn once an hour and tested for defect. What is the probability of finding a defect Note: a binomial pdistribution table must be used. Sample size is on far left signified byn (number of trials), and r =defects found,p = probability of finding the defect.
n = # trials
p = probability of a failure
Normal
1) All normal curves are bell-shaped with points of inflection at ± .
2) All normal curves are symmetric about the mean .
3) The area under an entire normal curve is 1.
4) All normal curves are positive for all x. That is, f(x) > 0 for all x.
5) The height of any normal curve is maximized at x = µ.
6) The shape of any normal curve depends on its mean and standard deviation .
Modern portfolio theory assumes returns of diversified asset portfolio follow a normal distribution
In operations management, process variations often are normally distributed.
µ is the mean and 2 is the variance of the distribution. The notation N(µ, 2 ) should be mentioned.
F
(1) F-distributions are generally skewed.
(2) The probability density function of an F random variable with r1 numerator degrees of freedom and r2 denominator degrees of freedom is:
f(w)=(r1/r2)r1/2[(r1+r2)/2]w(r1/2)1[r1/2][r2/2][1+(r1w/r2)](r1+r2)/2f(w)=(r1/r2)r1/2[(r1+r2)/2]w(r1/2)1[r1/2][r2/2][1+(r1w/r2)](r1+r2)/2
over the support w 0.
(3) The definition of an F-random variable:
F=U/r1V/r2F=U/r1V/r2
implies that if the distribution of W is F(r1, r2), then the distribution of 1/W is F(r2, r1).
The F-distribution arises from inferential statistics concerning population variances. More specifically, we use an F-distribution when we are studying the ratio of the variances of two normally distributed populations.
The F-distribution is not solely used to construct confidence intervals and test hypotheses about population variances. This type of distribution is also used in one factor analysis of variance (ANOVA). ANOVA is concerned with comparing the variation between several groups and variation within each group. To accomplish this we utilize a ratio of variances. This ratio of variances has the F-distribution. A somewhat complicated formula allows us to calculate an F-statistic as a test statistic.
F-distribution with r1 numerator degrees of freedom and r2 denominator degrees of freedom. We write F ~ F(r1,r2).
Exponential
Suppose that XX takes values in [0,)[0,). Then XX has the memoryless property if the conditional distribution of XsXs given X>sX>s is the same as the distribution of XX for every s[0,)s[0,). Equivalently,
P(X>t+sX>s)=P(X>t),s,t[0,)
XX has a continuous distribution and there exists r(0,)r(0,) such that the distribution function FF of XX is
F(t)=1ert,t[0,)
In the gamma experiment, set n=1n=1 so that the simulated random variable has an exponential distribution. Vary rr with the scroll bar and watch how the shape of the probability density function changes. For selected values of rr, run the experiment 1000 times and compare the empirical density function to the probability density function.
In the special distribution calculator, select the exponential distribution. Vary the scale parameter (which is 1/r1/r) and note the shape of the distribution/quantile function. For selected values of the parameter, compute a few values of the distribution function and the quantile function.
The exponential distribution can be used as a good approximate model for the time until the next phone call arrives.
Reliability theory and reliability engineering also make extensive use of the exponential distribution.
In hydrology, the exponential distribution is used to analyse extreme values of such variables as monthly and annual maximum values of daily rainfall and river discharge volumes.
Poisson
- It is a discrete distribution.
- Each occurrence is independent of the other occurrences.
- It describes discrete occurrences over an interval.
- The occurrences in each interval can range from zero to infinity.
- The mean number of occurrences must be constant throughout the experiment.
Estimating the number of car crashes in a city of a given size; in physiology,
Calculation of the probabilistic frequencies of different types of neurotransmitter secretions.