SMHS ProbabilityDistributions - Revision history

Dinov: /* Generating Probability Tables */

2016-03-18T18:44:37Z

‎Generating Probability Tables

Dinov: /* Generating Probability Tables */

2016-03-18T14:08:19Z

‎Generating Probability Tables

Dinov: /* Generating Probability Tables */

2016-03-18T14:06:35Z

‎Generating Probability Tables

Dinov: /* Generating Probability Tables */

2016-03-18T13:52:15Z

‎Generating Probability Tables

Dinov: /* Generating Probability Tables */

2016-03-18T13:51:52Z

‎Generating Probability Tables

Dinov: /* Applications */

2016-03-18T13:49:34Z

‎Applications

Dinov: /* MGF Properties */

2015-05-21T16:58:25Z

‎MGF Properties

Glenbrau: /* Problems */

2015-03-27T20:00:20Z

‎Problems

Glenbrau: /* Applications */

2015-03-27T19:55:12Z

‎Applications

Glenbrau: /* Applications */

2015-03-27T19:54:42Z

‎Applications

@@ Line 180: / Line 180: @@
   f_table
-  # lable rows and columns
+  # label rows and columns
   rownames(f_table) <- n2; colnames(f_table) <- n1

@@ Line 148: / Line 148: @@
 ===Generating Probability Tables===
-Once can use R (and many other programming languages) to generate probability tables like the [http://socr.umich.edu/Applets/index.html#Tables popular SOCR Probability Tables].
+Once can use R (and many other programming languages) to generate probability tables like the [http://socr.umich.edu/Applets/index.html#Tables popular SOCR Probability Tables]. You can also use the [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Java Applet] or the [http://www.distributome.org/V3/calc/FCalculator.html HTML5/JavaScript Webapp] interactive
 The following example generates one of the [http://socr.umich.edu/Applets/F_Table.html F distribution tables: $F(\alpha=0.001, df.num, df.deno)$]:

@@ Line 156: / Line 156: @@
   # Define the vectors storing the indices corresponding to numerator (n1) and denominator (n2, row)
-  # degrees of freedom for $F(\alpha, n1, n2)$. Note that Inf corresponds to $\infty$.
+  # degrees of freedom for $F(\alpha, n_1, n_2)$. Note that Inf corresponds to $\infty$.
   n1 <- c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 20, 24, 30, 40, 60, 120,Inf)
   n2 <- c(1:30, 40, 60, 120, Inf)

@@ Line 171: / Line 171: @@
   for (i in 1:length(n2)){
       for (j in 1:length(n1)){
-	f_table[i,j] <- qf(right_tail_p, n1[j], n2[i], lower.tail = FALSE)
+ 	f_table[i,j] <- qf(right_tail_p, n1[j], n2[i], lower.tail = FALSE)
+      }
+  }

@@ Line 177: / Line 177: @@
   # Print results
   f_table
-−
   # lable rows and columns
   rownames(f_table) <- n2; colnames(f_table) <- n1

← Older revision		Revision as of 13:49, 18 March 2016
Line 146:		Line 146:

	*[http://wiki.socr.umich.edu/index.php/SOCR_EduMaterials_Activities_NegativeBinomial This article] provides an example of Negative Binomial Experiment by SOCR. The goal is to provide a simulation demonstrating properties of the Negative Binomial ($k,p$) distribution. The applet facilitates the calculations of the Negative Binomial mass/density function, the moments and cumulative distribution function. It gives the specific steps of the experiment in SOCR and it allows users to learn about the variation of the distribution with changing parameters.		*[http://wiki.socr.umich.edu/index.php/SOCR_EduMaterials_Activities_NegativeBinomial This article] provides an example of Negative Binomial Experiment by SOCR. The goal is to provide a simulation demonstrating properties of the Negative Binomial ($k,p$) distribution. The applet facilitates the calculations of the Negative Binomial mass/density function, the moments and cumulative distribution function. It gives the specific steps of the experiment in SOCR and it allows users to learn about the variation of the distribution with changing parameters.
		+
		+	===Generating Probability Tables===
		+	Once can use R (and many other programming languages) to generate probability tables like the [http://socr.umich.edu/Applets/index.html#Tables popular SOCR Probability Tables].
		+
		+	The following example generates one of the [http://socr.umich.edu/Applets/F_Table.html F distribution tables: $F(\alpha=0.001, df.num, df.deno)$]:
		+
		+	# Define the right-tail probability of interest $\alpha=0.001$
		+	right_tail_p <- 0.001
		+
		+	# Define the vectors storing the indices corresponding to numerator (n1) and denominator (n2, row)
		+	# degrees of freedom for $F(\alpha, n1, n2)$. Note that Inf corresponds to $\infty$.
		+	n1 <- c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 20, 24, 30, 40, 60, 120,Inf)
		+	n2 <- c(1:30, 40, 60, 120, Inf)
		+
		+	# Define precision (4-decimal point accuracy)
		+	options(digits=4)
		+
		+	# Generate an empty matrix of critical f-values
		+	f_table <- matrix(ncol=length(n1), nrow=length(n2))
		+
		+	# Use the The F Distribution quantile function to fill in the matrix values in a nested 2-loop
		+	# Recall that the density (df), distribution function (pf), quantile function (qf) and random generation (rf) for the F distribution
		+
		+	for (i in 1:length(n2)){
		+	for (j in 1:length(n1)){
		+	f_table[i,j] <- qf(right_tail_p, n1[j], n2[i], lower.tail = FALSE)
		+	}
		+	}
		+
		+	# Print results
		+	f_table
		+
		+	# lable rows and columns
		+	rownames(f_table) <- n2; colnames(f_table) <- n1
		+
		+	# save results to a file
		+	write.table(f_table, file="C:\\User\\f_table.txt")

	===Software===		===Software===

@@ Line 128: / Line 128: @@
 : Hence, So, for a given MGF function, the power-series (Taylor) expansion of this function of $t$ yields the $r^{th}$ moment of hte random variable at the origin, as the coefficient, $E(X^k)$, of the term $\frac{t^k} {k!}$ in the expansion.
-* Moments about he origin may also be determined by differentiating the MGF:
+* Moments about the origin may also be determined by differentiating the MGF:
 $$\frac{d^k} {t^k} \{M_X(t) \} = \frac{d^k} {t^k} \{E(e^{tX}) \} = E \bigg [ \frac{d^k} {t^k} e^{tX} \bigg ] = E\bigg [ X^ke^{tX} \bigg ].$$

@@ Line 156: / Line 156: @@
 ===Problems===
 * If sampling distributions of sample means are examined for samples of size 1, 5, 10, 16 and 50, you will notice that as sample size increases, the shape of the sampling distribution appears more like that of the:
-: (a) normal distribution
+: (a) Normal distribution
-: (b) uniform distribution
+: (b) Uniform distribution
-: (c) population distribution
+: (c) Population distribution
-: (d) binomial distribution
+: (d) Binomial distribution
-* Which of the following statements best describes the effect on the Binomial Probability Model if the number of trials is held constant and the p(the probability of "success") increases?
+* Which of the following statements best describes the effect on the Binomial Probability Model if the number of trials has held constant and the $p$ (probability of "success") increases?
-: (a) None of these statements are true
+: (a) None of these statements are true.
-: (b) The mean and the standard deviation both increase
+: (b) The mean and the standard deviation both increase.
-: (c) The mean decreases and the standard deviation increases
+: (c) The mean decreases and the standard deviation increases.
-: (d) The mean increases and the standard deviation decreases
+: (d) The mean increases and the standard deviation decreases.
-: (e) The mean and standard deviation both decrease
+: (e) The mean and standard deviation both decrease.
 * Suppose you draw one card from a standard deck three times, with replacement. What is the probability that you get spades all three times? Choose one answer.
@@ Line 174: / Line 174: @@
 : (d) 0.021
-* Suppose the number of cars that enter a parking lot in an hour is a Poisson random variable, and suppose that P(X=0)=0.05. Determine the variance of X.
+* Suppose the number of cars that enter a parking lot in an hour is a Poisson random variable, and suppose that $P(X=0)=0.05$. Determine the variance of $X$.
 : (a) 0.349
 : (b) 3.232
@@ Line 181: / Line 181: @@
 * A researcher converts 100 lung capacity measurements to z-scores. The lung capacity measurements do not follow a normal distribution. What can we say about the standard deviation of the 100 z-scores?
-: (a) It depends on the standard deviation of the raw scores
+: (a) It depends on the standard deviation of the raw scores.
-: (b) It equals 1
+: (b) It equals 1.
-: (c) It equals 100
+: (c) It equals 100.
-: (d) It must always be less than the standard deviation of the raw scores
+: (d) It must always be less than the standard deviation of the raw scores.
-: (e) It depends on the shape of the raw score distribution
+: (e) It depends on the shape of the raw score distribution.
-* Among first year students at a certain university, scores on the verbal SAT follow the normal curve. The average is around 500 and the SD is about 100. Tatiana took the SAT, and placed at the 85% percentile. What was her verbal SAT score?
+* Among first year students at a certain university, scores on the verbal SAT follow the normal curve. The average is around 500 and the $SD$ is about 100. Tatiana took the SAT, and placed at the 85% percentile. What was her verbal SAT score?
 : (a) 604
 : (b) 560
@@ Line 193: / Line 193: @@
 : (d) 403
-* Consider a random sample 100 orc soldiers and found the mean and the standard deviation to be 200lbs and and 20lbs respectively. He can be 68% confident that the mean weight in the population of orc soldiers is between
+* Consider a random sample of 100 orc soldiers in which the mean and the standard deviation are 200 lbs. and and 20 lbs., respectively. We can be 68% confident that the mean weight in the population of orc soldiers is between
 : (a) 196 to 204 lbs
 : (b) 198 to 202 lbs
@@ Line 211: / Line 211: @@
 : (d) 62.44 inches
-* The settlement (in cm) of a structure shown in the following figure may be evaluated from S = 0.3A + 0.2B + 0.1C,
+* The settlement (in cm.) of a structure shown in the following figure may be evaluated from S = 0.3A + 0.2B + 0.1C,
 <center>[[Image:SMHS_ProbabDist_Fig3.png]]</center>
-: where A, B, and C are respectively the thickness (in m) of the three layers of soil as shown. Suppose A, B, and C are modeled as independent normal random variables as: $A \sim N(5,1)$, $B \sim N(8,2)$, $C \sim N(7,1)$,
+: Where A, B, and C are respectively the thickness (in m) of the three layers of soil as shown. Suppose A, B, and C are modeled as independent normal random variables as: $A \sim N(5,1)$, $B \sim N(8,2)$, $C \sim N(7,1)$,
 : (a) Determine the probability that the settlement will exceed 4 cm.
 : (b) If the total thickness of the three layers is known exactly as 20 m; and furthermore, thicknesses A and B are correlated with correlation coefficient equal to 0.5, determine the probability that the settlement will exceed 4 cm.
-* Suppose that the distribution of X in the population is strongly skewed to the left. If you took 200 independent and random samples of size 3 from this population, calculated the mean for each of the 200 samples, and drew the distribution of the sample means, what would the sampling distribution of the means look like?
+* Suppose that the distribution of $X$ in the population is strongly skewed to the left. If you took 200 independent and random samples of size 3 from this population, calculated the mean for each of the 200 samples and drew the distribution of the sample means, what would the sampling distribution of the means look like?
 : (a) It will be perfectly normal and the mean will be equal to the median.
 : (b) It will be close to the normal and the mean will be close to the median.
@@ Line 224: / Line 224: @@
 * A polling agency has been hired to predict the proportion of voters who favor a certain candidate. The polling agency picks a random sample of 1000 voters of which 400 indicate that they favor the candidate. If they increase the sample size to 2000, how does the standard error change?
-: (a) The standard error will decrease by one-fourth
+: (a) The standard error will decrease by one-fourth.
-: (b) The standard error will not change; the margin of error changes
+: (b) The standard error will not change; the margin of error changes.
-: (c) Since the sample size is doubled, the standard error will be halved
+: (c) Since the sample size is doubled, the standard error will be halved.
-: (d) The standard error will decrease not by a factor of 1/2 but by the square of root of 1/2
+: (d) The standard error will decrease not by a factor of 1/2 but by the square of root of 1/2.
 * The probability of winning a certain instant scratch-n-win game is 0.02. You play the game 80 times. Find the probability that you win 3 times.
@@ Line 235: / Line 235: @@
 : (d) 0.2391
-* After firing 1000 boxes of ammunition, a certain handgun jamed according to a Poisson distribution with a mean of 0.4 per box of ammunition. Approximate the probability that more than 350 boxes of ammunition contain some that is jammed.
+* After firing 1,000 boxes of ammunition, a certain handgun jamed according to a Poisson distribution with a mean of 0.4 per box of ammunition. Approximate the probability that more than 350 boxes of ammunition contain some that is jammed.
 : (a) .0032
 : (b) .0012

@@ Line 145: / Line 145: @@
 *[http://www.mdm.com/articles/28757-the-case-for-proactive-inside-sales?v=preview  The article] examines how a proactive inside sales force can be critical to serving mid-market and small customers as part of a broader multichannel strategy. It also includes steps for initiating an effective program.
-*[http://wiki.socr.umich.edu/index.php/SOCR_EduMaterials_Activities_NegativeBinomial This article] provides an example of Negative Binomial Experiment by SOCR. The goal of this experiment is to provide a simulation demonstrating properties of the Negative Binomial ($k,p$) distribution. The applet facilitates the calculations of the Negative Binomial mass/density function, the moments and cumulative distribution function. It gives the specific steps of the experiment in SOCR and it allows users to learn about the variation of the distribution with changing parameters.
+*[http://wiki.socr.umich.edu/index.php/SOCR_EduMaterials_Activities_NegativeBinomial This article] provides an example of Negative Binomial Experiment by SOCR. The goal is to provide a simulation demonstrating properties of the Negative Binomial ($k,p$) distribution. The applet facilitates the calculations of the Negative Binomial mass/density function, the moments and cumulative distribution function. It gives the specific steps of the experiment in SOCR and it allows users to learn about the variation of the distribution with changing parameters.
 ===Software===

@@ Line 143: / Line 143: @@
 ===Applications===
-*[http://www.mdm.com/articles/28757-the-case-for-proactive-inside-sales?v=preview  The article] examined how a proactive inside sales force can be critical to serving mid-market and small customers as part of a broader multichannel strategy and in included steps for initiating an effective program.
+*[http://www.mdm.com/articles/28757-the-case-for-proactive-inside-sales?v=preview  The article] examines how a proactive inside sales force can be critical to serving mid-market and small customers as part of a broader multichannel strategy. It also includes steps for initiating an effective program.
-*[http://wiki.socr.umich.edu/index.php/SOCR_EduMaterials_Activities_NegativeBinomial This article] provides an example of Negative Binomial Experiment by SOCR. The goal of this experiment is to provide a simulation demonstrating properties of the Negative Binomial(k,p) distribution. The applet facilitates the calculations of the Negative Binomial mass/density function, the moments and cumulative distribution function. It gives the specific steps of the experiment in SOCR and it allows users to learn about the variation of the distribution with changing parameters.
+*[http://wiki.socr.umich.edu/index.php/SOCR_EduMaterials_Activities_NegativeBinomial This article] provides an example of Negative Binomial Experiment by SOCR. The goal of this experiment is to provide a simulation demonstrating properties of the Negative Binomial ($k,p$) distribution. The applet facilitates the calculations of the Negative Binomial mass/density function, the moments and cumulative distribution function. It gives the specific steps of the experiment in SOCR and it allows users to learn about the variation of the distribution with changing parameters.
 ===Software===