Lab 5: Probability density and cumulative distri- butions 1 In trod ucti on The purpose of this lab is to become familiar with the concepts of• probability density, • cumulative distribution, • mean, and • median. In particular, we will be studying the empirical data from some Bernoulli trials, which should fit a binomial distribution. We will plot the probability density and cumulative distribution as bar graphs, and then compute the mean and median from these bar graphs. The mean ¯ x of a random variable whose possible values x i have probabilities p(x i ) is defined by ¯ x = i x i p(x i ). The median m of an (unordered) list of numbers is the middle number after ordering. For instance, if the ordered list is 1, 2, 2, 2, 3, 3, 4 then the middle number (media n) is 2. Thus, the median of the related random variable is the first x for which the cumulative distribution function exceeds (or is equal to) 0 .5. Convin ce your sel f that this is so! (In cases of an even number of data points the convention is to use the average of the middle two, but this does not occur in this lab.) [ Introduction Problem Set 1 Problem Set 2 Problem Set 3 Problem Set 4 Problem Set 5 Previous Page Next Page Exit ]
Lab 5: Probability density and cumulative distri-butions
1 Introduction
The purpose of this lab is to become familiar with the concepts of
• probability density,
• cumulative distribution,
• mean, and
• median.
In particular, we will be studying the empirical data from some Bernoulli trials, whichshould fit a binomial distribution. We will plot the probability density and cumulativedistribution as bar graphs, and then compute the mean and median from these bargraphs.
The mean x̄ of a random variable whose possible values xi have probabilities p(xi) isdefined by
x̄ =i
xi p(xi).
The median m of an (unordered) list of numbers is the middle number after ordering.For instance, if the ordered list is 1, 2, 2, 2, 3, 3, 4 then the middle number (median) is 2.Thus, the median of the related random variable is the first x for which the cumulativedistribution function exceeds (or is equal to) 0.5. Convince yourself that this is so!(In cases of an even number of data points the convention is to use the average of themiddle two, but this does not occur in this lab.)
Given below is the distribution of number of heads obtained by a group of people inan experiment in which each person tossed a fair coin ten times.Let:x = number of heads obtained out of ten tosses.N (x)=number of people who got x heads.
• Use the above data to plot a bar graph of p(x), i.e., the empirical probabilitydistribution of obtaining x heads in 10 tosses. Center the bars at the integers. (Seenote at the bottom for plotting.)
• Plot the cumulative distribution (of obtaining up to x heads) for the empirical data.Technically this should be a step function, but step functions are difficult to displayusing Mathsheet, so we suggest using a bar graph.
• Use the spreadsheet to calculate the expected number of heads based on the sameempirical data.
• Find the median number of heads of the empirical data.
• Finally, plot a bar graph of the theoretical probability distribution in the same way.
• Submit a graph of these three bar graphs and write the values you found for themean and the median of the empirical data on the bottom of the page. Also, pleaseexplain how you computed the median.
Table 1 Problem Set 1: Of 124 peopleN (x) tossed x heads
Note: It is somewhat tricky to display all three of these bar graphs without having
them cover eachother. One possible solution is to plot the cumulative distribution first(filled) with the empirical probability distribution filled on top. Finally, the theoreticalprobability distribution can be plotted over everything, but with the fill setting off sothat the other graphs show through.
Figure 1 Solution for problem set 1. The median number of heads is 5,
(first x for which the cumulative distribution function exceeds 0.5) and theexpected number of heads (also called mean number of heads) is calculatedby the spreadsheet as x̄ = 5.1.
Here’s how the data could be organized: Column 1 contains the number x of headsfrom 1 to 10. Column 2 contains the number of people who got x heads. Column 3contains the theoretically expected number of people who got x heads. Column 4 totals
up the size of the group of people. (There were 124 people in all). The numbers in
column 5 are then the fraction N (x)124 of the group that got a given number of heads.
This is the probability p(x) of getting x heads. Column 6 computes the cumulativefunction F (x). Column 7 computes xp(x) and column 8 then sums these up to get themean value of x for the probability density.We see that the mean value is x̄ = 5.1.
Figure 1 then shows the graphs of p(x) and F (x) and the theoretical value on the samegraph.From this graph, we can read off the median value. We locate the x at which thecumulative distribution function (blue curve) exceeds the value 0.5 for the first time.This is at m = 5.
Given below is the distribution of number of heads obtained by a group of people inan experiment in which each person tossed a fair coin ten times.Let:x = number of heads obtained out of ten tosses.
N (x)=number of people who got x heads.
x N (x)
0 0
1 1
2 3
3 134 16
5 14
6 20
7 9
8 5
9 1
10 0
Table 2 Problem Set 2: Of 82 peopleN (x) tossed x heads
• Use the above data to plot a bar graph of p(x), i.e., the empirical probabilitydistribution of obtaining x heads in 10 tosses. Center the bars at the integers. (Seenote at the bottom for plotting.)
• Plot the cumulative distribution (of obtaining up to x heads) for the empirical data.Technically this should be a step function, but step functions are difficult to display
using Mathsheet, so we suggest using a bar graph.• Use the spreadsheet to calculate the expected number of heads based on the same
empirical data.
• Find the median number of heads of the empirical data.
• Finally, plot a bar graph of the theoretical probability distribution in the same way.
• Submit a graph of these three bar graphs and write the values you found for themean and the median of the empirical data on the bottom of the page. Also, pleaseexplain how you computed the median.
Note: It is somewhat tricky to display all three of these bar graphs without havingthem cover eachother. One possible solution is to plot the cumulative distribution first(filled) with the empirical probability distribution filled on top. Finally, the theoretical
probability distribution can be plotted over everything, but with the fill setting off sothat the other graphs show through.
Figure 2 Solution for problem set 2. The median number of heads is 5,
(first x for which the cumulative distribution function exceeds 0.5) and theexpected number of heads (also called mean number of heads) is calculatedby the spreadsheet as x̄ = 5.02.
Here’s how the data could be organized: Column 1 contains the number x of headsfrom 1 to 10. Column 2 contains the number of people who got x heads. Column 3contains the theoretically expected number of people who got x heads. Column 4
totals up the size of the group of people. (There were 82 people in all). The numbers
in column 5 are then the fraction N (x)82 of the group that got a given number of heads.
This is the probability p(x) of getting x heads. Column 6 computes the cumulativefunction F (x). Column 7 computes xp(x) and column 8 then sums these up to get themean value of x for the probability density.We see that the mean value is x̄ = 5.02.
Figure 2 then shows the graphs of p(x) and F (x) and the theoretical value on the samegraph.From this graph, we can read off the median value. We locate the x at which thecumulative distribution function (blue curve) exceeds the value 0.5 for the first time.This is at m = 5.
Given below is the distribution of number of heads obtained by a group of people inan experiment in which each person tossed a fair coin ten times.Let:x = number of heads obtained out of ten tosses.
N (x)=number of people who got x heads.
x N (x)
0 0
1 1
2 9
3 174 25
5 37
6 20
7 14
8 6
9 1
10 0
Table 3 Problem Set 3: Of 130 peopleN (x) tossed x heads
• Use the above data to plot a bar graph of p(x), i.e., the empirical probabilitydistribution of obtaining x heads in 10 tosses. Center the bars at the integers. (Seenote at the bottom for plotting.)
• Plot the cumulative distribution (of obtaining up to x heads) for the empirical data.Technically this should be a step function, but step functions are difficult to display
using Mathsheet, so we suggest using a bar graph.• Use the spreadsheet to calculate the expected number of heads based on the same
empirical data.
• Find the median number of heads of the empirical data.
• Finally, plot a bar graph of the theoretical probability distribution in the same way.
• Submit a graph of these three bar graphs and write the values you found for themean and the median of the empirical data on the bottom of the page. Also, pleaseexplain how you computed the median.
Note: It is somewhat tricky to display all three of these bar graphs without havingthem cover eachother. One possible solution is to plot the cumulative distribution first(filled) with the empirical probability distribution filled on top. Finally, the theoretical
probability distribution can be plotted over everything, but with the fill setting off sothat the other graphs show through.
Figure 3 Solution for problem set 3. The median number of heads is 5,
(first x for which the cumulative distribution function exceeds 0.5) and theexpected number of heads (also called mean number of heads) is calculatedby the spreadsheet as x̄ = 4.85.
Here’s how the data could be organized: Column 1 contains the number x of headsfrom 1 to 10. Column 2 contains the number of people who got x heads. Column 3contains the theoretically expected number of people who got x heads. Column 4 totals
up the size of the group of people. (There were 130 people in all). The numbers in
column 5 are then the fraction N (x)130 of the group that got a given number of heads.
This is the probability p(x) of getting x heads. Column 6 computes the cumulativefunction F (x). Column 7 computes xp(x) and column 8 then sums these up to get themean value of x for the probability density.We see that the mean value is x̄ = 4.85.
Figure 3 then shows the graphs of p(x) and F (x) and the theoretical value on the samegraph.From this graph, we can read off the median value. We locate the x at which thecumulative distribution function (blue curve) exceeds the value 0.5 for the first time.This is at m = 5.
Given below is the distribution of number of heads obtained by a group of people inan experiment in which each person tossed a fair coin ten times.Let:x = number of heads obtained out of ten tosses.N
(x
)=number of people who gotx
heads.
x N (x)
0 0
1 0
2 1
3 8
4 35
5 17
6 20
7 14
8 3
9 1
10 0
Table 4 Problem Set 4: Of 99 peopleN (x) tossed x heads
[
Introduction
ProblemSet 1
ProblemSet 2
ProblemSet 3
ProblemSet 4
ProblemSet 5
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Exit ]
U th b d t t l t b h f ( ) i th i i l b bilit I d i
• Use the above data to plot a bar graph of p(x), i.e., the empirical probabilitydistribution of obtaining x heads in 10 tosses. Center the bars at the integers. (Seenote at the bottom for plotting.)
• Plot the cumulative distribution (of obtaining up to x heads) for the empirical data.Technically this should be a step function, but step functions are difficult to displayusing Mathsheet, so we suggest using a bar graph.
• Use the spreadsheet to calculate the expected number of heads based on the sameempirical data.
• Find the median number of heads of the empirical data.
• Finally, plot a bar graph of the theoretical probability distribution in the same way.
• Submit a graph of these three bar graphs and write the values you found for themean and the median of the empirical data on the bottom of the page. Also, pleaseexplain how you computed the median.
Note: It is somewhat tricky to display all three of these bar graphs without havingthem cover eachother. One possible solution is to plot the cumulative distribution first(filled) with the empirical probability distribution filled on top. Finally, the theoretical
probability distribution can be plotted over everything, but with the fill setting off sothat the other graphs show through.
Figure 4 Solution for problem set 4. The median number of heads is 5,
(first x for which the cumulative distribution function exceeds 0.5) and theexpected number of heads (also called mean number of heads) is calculatedby the spreadsheet as x̄ = 5.07.
Here’s how the data could be organized: Column 1 contains the number x of headsfrom 1 to 10. Column 2 contains the number of people who got x heads. Column 3contains the theoretically expected number of people who got x heads. Column 4
[
Introduction
ProblemSet 1
ProblemSet 2
ProblemSet 3
ProblemSet 4
ProblemSet 5
Previous Page
Next Page
Exit ]
totals up the size of the group of people (There were 99 people in all) The numbers Int od ction
totals up the size of the group of people. (There were 99 people in all). The numbers
in column 5 are then the fraction N (x)99 of the group that got a given number of heads.
This is the probability p(x) of getting x heads. Column 6 computes the cumulativefunction F (x). Column 7 computes xp(x) and column 8 then sums these up to get themean value of x for the probability density.We see that the mean value is x̄ = 5.07.
Figure 4 then shows the graphs of p(x) and F (x) and the theoretical value on the samegraph.From this graph, we can read off the median value. We locate the x at which thecumulative distribution function (blue curve) exceeds the value 0.5 for the first time.This is at m = 5.
Given below is the distribution of number of heads obtained by a group of people inan experiment in which each person tossed a fair coin ten times.Let:x = number of heads obtained out of ten tosses.N (x)=number of people who got x heads.
x N (x)
0 0
1 2
2 1
3 17
4 15
5 31
6 28
7 10
8 2
9 1
10 0
Table 5 Problem Set 5: Of 107 peopleN (x) tossed x heads
[
Introduction
ProblemSet 1
ProblemSet 2
ProblemSet 3
ProblemSet 4
ProblemSet 5
Previous Page
Next Page
Exit]
• Use the above data to plot a bar graph of p(x) i e the empirical probability Introduction
• Use the above data to plot a bar graph of p(x), i.e., the empirical probabilitydistribution of obtaining x heads in 10 tosses. Center the bars at the integers. (Seenote at the bottom for plotting.)
• Plot the cumulative distribution (of obtaining up to x heads) for the empirical data.Technically this should be a step function, but step functions are difficult to displayusing Mathsheet, so we suggest using a bar graph.
• Use the spreadsheet to calculate the expected number of heads based on the sameempirical data.
• Find the median number of heads of the empirical data.
• Finally, plot a bar graph of the theoretical probability distribution in the same way.
• Submit a graph of these three bar graphs and write the values you found for themean and the median of the empirical data on the bottom of the page. Also, pleaseexplain how you computed the median.
Note: It is somewhat tricky to display all three of these bar graphs without havingthem cover eachother. One possible solution is to plot the cumulative distribution first(filled) with the empirical probability distribution filled on top. Finally, the theoretical
probability distribution can be plotted over everything, but with the fill setting off sothat the other graphs show through.
Figure 5 Solution for problem set 5. The median number of heads is 5,
(first x for which the cumulative distribution function exceeds 0.5) and theexpected number of heads (also called mean number of heads) is calculatedby the spreadsheet as x̄ = 4.98.
Here’s how the data could be organized: Column 1 contains the number x of headsfrom 1 to 10. Column 2 contains the number of people who got x heads. Column 3contains the theoretically expected number of people who got x heads. Column 4 totals
[
Introduction
ProblemSet 1
ProblemSet 2
ProblemSet 3
ProblemSet 4
ProblemSet 5
Previous Page
Next Page
Exit]
up the size of the group of people. (There were 107 people in all). The numbers in Introduction
up the size of the group of people. (There were 107 people in all). The numbers in
column 5 are then the fraction N (x)107 of the group that got a given number of heads.
This is the probability p(x) of getting x heads. Column 6 computes the cumulativefunction F (x). Column 7 computes xp(x) and column 8 then sums these up to get themean value of x for the probability density.We see that the mean value is x̄ = 4.98.
Figure 5 then shows the graphs of p(x) and F (x) and the theoretical value on the samegraph.From this graph, we can read off the median value. We locate the x at which thecumulative distribution function (blue curve) exceeds the value 0.5 for the first time.This is at m = 5.
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