The PMF is one way to describe the distribution of a discrete random variable. As we will see later on, PMF cannot be defined for continuous random variables. The cumulative distribution function (CDF) of a random variable is another method to describe the distribution of random variables. The advantage of the CDF is that it can be defined for any kind of random variable (discrete, continuous, and mixed).
Definition
The cumulative distribution function (CDF) of random variable $X$ is defined as $$F_X(x) = P(X leq x), textrm{ for all }x in mathbb{R}.$$
The cumulative distribution function (CDF) of random variable $X$ is defined as $$F_X(x) = P(X leq x), textrm{ for all }x in mathbb{R}.$$
Note that the subscript $X$ indicates that this is the CDF of the random variable $X$. Also, note that the CDF is defined for all $x in mathbb{R}$. Let us look at an example.
With the Wolfram Player app, read and dynamically interact with Wolfram Notebooks and the Computable Document Format (CDF) through a tactile and responsive interface—all running locally on iOS and sideloading from the Wolfram Cloud, email, Dropbox, Google Drive, iTunes or elsewhere. Contextual Information Of CDF Traces: Very Important( Without these logs cannot be seen) i. Time at which we started CDF Tracing on all DDCs: DD/MM/YYYY HH:MM:SS ii. Time at which we started Remote CDF Tracing on VDA: DD/MM/YYYY HH:MM:SS iii. The CDF ID, returned from a previous call to CDFOPEN or CDFCREATE. A string containing the name of the attribute or the attribute number to be written. The entry number. If the attribute is variable in scope, this is either the name or number of the variable the attribute is to be associated with.
Example
I toss a coin twice. Let $X$ be the number of observed heads. Find the CDF of $X$.
- Solution
- Note that here $X sim Binomial (2, frac{1}{2})$. The range of $X$ is $R_X={0,1,2}$ and its PMF is given by $$P_X(0)=P(X=0)=frac{1}{4},$$ $$P_X(1) =P(X=1)=frac{1}{2},$$ $$P_X(2)=P(X=2)=frac{1}{4}.$$ To find the CDF, we argue as follows. First, note that if $x < 0$, then $$F_X(x)=P(X leq x)=0, textrm{ for } x < 0.$$ Next, if $xgeq 2$, $$F_X(x)=P(X leq x)=1, textrm{ for } xgeq 2.$$ Next, if $0 leq x < 1$, $$F_X(x)=P(X leq x)=P(X=0)=frac{1}{4}, textrm{ for } 0 leq x < 1.$$ Finally, if $1 leq x < 2$, $$F_X(x)=P(X leq x)=P(X=0)+P(X=1)=frac{1}{4}+frac{1}{2}=frac{3}{4}, textrm{ for } 1 leq x < 2.$$ Thus, to summarize, we have begin{equation} nonumber F_X(x) = left{ begin{array}{l l} 0 & quad text{for } x < 0 frac{1}{4} & quad text{for } 0 leq x < 1 frac{3}{4} & quad text{for } 1 leq x < 2 1 & quad text{for } x geq 2 end{array} right. end{equation} Note that when you are asked to find the CDF of a random variable, you need to find the function for the entire real line. Also, for discrete random variables, we must be careful when to use '$ < $' or '$leq$'. Figure 3.3 shows the graph of $F_X(x)$. Note that the CDF is flat between the points in $R_X$ and jumps at each value in the range. The size of the jump at each point is equal to the probability at that point. For, example, at point $x=1$, the CDF jumps from $frac{1}{4}$ to $frac{3}{4}$. The size of the jump here is $frac{3}{4}-frac{1}{4}=frac{1}{2}$ which is equal to $P_X(1)$. Also, note that the open and closed circles at point $x=1$ indicate that $F_X(1)=frac{3}{4}$ and not $frac{1}{4}$.
- Note that here $X sim Binomial (2, frac{1}{2})$. The range of $X$ is $R_X={0,1,2}$ and its PMF is given by $$P_X(0)=P(X=0)=frac{1}{4},$$ $$P_X(1) =P(X=1)=frac{1}{2},$$ $$P_X(2)=P(X=2)=frac{1}{4}.$$ To find the CDF, we argue as follows. First, note that if $x < 0$, then $$F_X(x)=P(X leq x)=0, textrm{ for } x < 0.$$ Next, if $xgeq 2$, $$F_X(x)=P(X leq x)=1, textrm{ for } xgeq 2.$$ Next, if $0 leq x < 1$, $$F_X(x)=P(X leq x)=P(X=0)=frac{1}{4}, textrm{ for } 0 leq x < 1.$$ Finally, if $1 leq x < 2$, $$F_X(x)=P(X leq x)=P(X=0)+P(X=1)=frac{1}{4}+frac{1}{2}=frac{3}{4}, textrm{ for } 1 leq x < 2.$$ Thus, to summarize, we have begin{equation} nonumber F_X(x) = left{ begin{array}{l l} 0 & quad text{for } x < 0 frac{1}{4} & quad text{for } 0 leq x < 1 frac{3}{4} & quad text{for } 1 leq x < 2 1 & quad text{for } x geq 2 end{array} right. end{equation}
In general, let $X$ be a discrete random variable with range $R_X={x_1,x_2,x_3,..}$, such that $x_1 < x_2 < x_3 < ..$ Here, for simplicity, we assume that the range $R_X$ is bounded from below, i.e., $x_1$ is the smallest value in $R_X$. If this is not the case then $F_X(x)$ approaches zero as $x rightarrow -infty$ rather than hitting zero. Figure 3.4 shows the general form of the CDF, $F_X(x)$, for such a random variable. We see that the CDF is in the form of a staircase. In particular, note that the CDF starts at $0$; i.e.,$F_X(-infty)=0$. Then, it jumps at each point in the range. In particular, the CDF stays flat between $x_k$ and $x_{k+1}$, so we can write $$F_X(x)=F_X(x_k), textrm{ for }x_k leq x < x_{k+1}.$$ Set a light 3d studio keygen download.
The CDF jumps at each $x_k$. In particular, we can write $$F_X(x_k)-F_X(x_k-epsilon)=P_X(x_k), textrm{ For $epsilon>0$ small enough.}$$ Thus, the CDF is always a non-decreasing function, i.e., if $y geq x$ then $F_X(y)geq F_X(x)$. Finally, the CDF approaches $1$ as $x$ becomes large. We can write $$lim_{x rightarrow infty} F_X(x)=1.$$ Visual modflow license cracked.
Note that the CDF completely describes the distribution of a discrete random variable. In particular, we can find the PMF values by looking at the values of the jumps in the CDF function. Also, if we have the PMF, we can find the CDF from it. In particular, if $R_X={x_1,x_2,x_3,..}$, we can write $$F_X(x)=sum_{x_k leq x} P_X(x_k).$$ Now, let us prove a useful formula.
For all $a leq b$, we have $$hspace{50pt} P(a < X leq b)=F_X(b)-F_X(a) hspace{80pt} (3.1)$$
To see this, note that for $a leq b$ we have $$P(X leq b)=P(X leq a) + P(a < X leq b).$$ Thus, $$F_X(b)=F_X(a) + P(a < X leq b).$$ Again, pay attention to the use of '$ < $' and '$leq$' as they could make a difference in the case of discrete random variables. We will see later that Equation 3.1 is true for all types of random variables (discrete, continuous, and mixed). Note that the CDF gives us $P(X leq x)$. To find $P(X < x)$, for a discrete random variable, we can simply write $$P(X < x)=P(X leq x)-P(X=x)=F_X(x)-P_X(x).$$
Example
Let $X$ be a discrete random variable with range $R_X={1,2,3,..}$. Suppose the PMF of $X$ is given by $$P_X(k)=frac{1}{2^k} textrm{ for } k=1,2,3,..$$
- Find and plot the CDF of $X$, $F_X(x)$.
- Find $P(2 < X leq 5)$.
- Find $P(X > 4)$.
- Solution
- First, note that this is a valid PMF. In particular, $$sum_{k=1}^{infty} P_X(k)=sum_{k=1}^{infty} frac{1}{2^k}=1 textrm{ (geometric sum)}$$
- To find the CDF, note that
$textrm{For } x < 1,$ $F_X(x)=0$. $textrm{For } 1leq x < 2,$ $F_X(x)=P_X(1)=frac{1}{2}$. $textrm{For } 2leq x < 3,$ $F_X(x)=P_X(1)+P_X(2)=frac{1}{2}+ frac{1}{4}=frac{3}{4}$.
In general we have $$textrm{For } 0 < k leq x < k+1,$$ $$F_X(x) =P_X(1)+P_X(2)+..+P_X(k)$$ $$=frac{1}{2}+ frac{1}{4}+..+frac{1}{2^k}=frac{2^k-1}{2^k}.$$Figure 3.5 shows the CDF of $X$. - To find $P(2 < X leq 5)$, we can write $$P(2 < X leq 5)=F_X(5)-F_X(2)=frac{31}{32}-frac{3}{4}=frac{7}{32}.$$ Or equivalently, we can write $$P(2 < X leq 5)=P_X(3)+P_X(4)+P_X(5)=frac{1}{8}+frac{1}{16}+frac{1}{32}=frac{7}{32},$$ which gives the same answer.
- To find $P(X > 4)$, we can write $$P(X > 4)=1-P(X leq 4)=1-F_X(4)=1-frac{15}{16}=frac{1}{16}.$$
- To find the CDF, note that
- First, note that this is a valid PMF. In particular, $$sum_{k=1}^{infty} P_X(k)=sum_{k=1}^{infty} frac{1}{2^k}=1 textrm{ (geometric sum)}$$
Language
CDFControl.zip
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Applicable Products
- XenDesktop
- XenApp
- Receiver for Windows
- Citrix Workspace App
- Citrix Virtual Apps and Desktops
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CDFControl v3.8.1.27
Created on: Mar 8, 2007
Updated on: Nov 18, 2019
Description
CDFControl is an event tracing tool geared towards capturing Citrix Diagnostic Facility (CDF) trace messages that are output from the various Citrix tracing providers.
New in this Version
- Use TLS 1.2 to access the Citrix Symbol Server.
Changes in Previous Versions
- Added support for saving TMF files downloaded during parsing to the local cache path at C:ProgramDataCitrixCDFControlTMFCache. This approach accelerates trace parsing.
- Updated .Net Framework from Version 2.0 to 4.5.1.
- Fixed an issue where CDFControl cannot find the modules of Citrix Workspace app.
- Accelerated trace parsing for .NET programming.
- Fixed an issue where CDFControl reports the error 'TMF file not found for AOT traces.'
- Fixed an issue where the LonglongX formatting type is not indexed in TMF caches.
- Added support for multithreading and multiprocessing.
- Fixed an issue where Parse Trace to File cannot always parse all lines.
- Added CDF metadata support for .NET trace in the Comments field.
- Fixed an issue where a blank filter might cause a null pointer exception.
- Fixed an issue where the CPU field is always 0.
- Fixed an issue where some PVS traces cannot be parsed.
- AOT trace parsing.
- AOT trace highlighting.
- Added a limitation on the number of files of multiple sequential logs captured.
- Changed the color for error highlighting from black on red to yellow on red.
- Added new trace modules for Citrix Receiver and Citrix Workspace app.
- Fixed a crash issue that occurs when parsing Receiver AOT traces.
- Enhanced search to all TMF search paths for certain GUIDs to trasverse the TMF file for all platforms.
- Added a setting to highlight errors and warnings automatically.
- Added a setting to enable PVS-specific.NET trace parsing format.
- Made the content of the trace class column more concise.
- Added missing modules for the local app access and printing features.
- Added support for parsing traces by using the time zone of the machine being traced.
- Introduced a warning dialog when you start tracing without selecting any module.
- Changed the default configuration to multiple sequential modes.
- Added the session tree of the DLL dependency list for processes.
- Added support for PVS.NET trace output with detailed columns.
- Signed binary files with SHA-256 to improve security.
- Changed the directory timestamp to 24-hour format.
- Introduced support for modifying and inserting custom trace messages in the Remote tracing dialog.
- Enhanced remote tracing support on Windows 10.
- Solved User Access Control (UAC) issues on Windows 7 and Windows 10.
- Introduced support to help parse multiple trace files (maximum 63 files) to a single output file. The trace messages are automatically sorted by message timestamp.
- Added a time filter option to help parse only trace messages in a specific time period.
- Moved to a new Citrix public symbol server since Version 3.0.2.0.
- Enhanced filtering to include filtering by column header, multiple rules, parsing filter, and more. Use the column header filter to filter the tracing from the column header; use multiple filtering rules to the current log at the same time; the parsing filter supports filtering by regular expression in real-time tracing and ETL parsing.
- Enhanced overall performance, which is mainly reflected in lower memory consumption and faster parsing speeds.
- Introduced a module filter that allows you to search for specific modules by binary or module name.
- Added support for inserting custom trace statements to a running trace to mark the occurrence of specific events.
- Added a comment column for traces. Use this feature to add your findings to the comments column, save them to a CSV file, and share.
- Introduced session/process information support for remote tracing.
- Introduced support for HPC (High Performance Client) tracing. CDFControl supports collecting HPC tracing logs on Windows machines with Citrix Receiver 4.1.100 or later installed. Internal users can view real-time HPC tracing logs. The original HPC tracing pattern was not changed. You can still enable the original HPC tracing.
- Enabled highlighting. Use this feature to highlight rows in the viewer to distinguish them from others.
- Introduced support for custom column display. Use this feature to decide which columns you want to display or hide.
- Introduced an Add Filter Logic function for the multi-rules filter. Use this feature to define your own filter logic for multiple rules rather than the fixed AND logic of earlier versions.
Prerequisites
- CDFControl requires Windows 2000 or later with .NET 2.0 or later installed.
- Run the tool as a user with administrator privileges for all features to be functional.
- On Microsoft Windows Vista and Windows 2008, with UAC, only trace capture requires elevation. For more information, see the user guide (CDFControl Menu > Help).
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To install CDFControl, copy the CDFControl.exe executable file to the desired location. For Windows 8 and later, you must copy CDFControl.exe.config as well. No installer is required.
The CDFControl Version 2 includes the following for both 32-bit and 64-bit versions:
- CDFControl.exe
(Can be run standalone) - CDFControl.exe.config
(Necessary only for Windows 8 and later)
The first time you launch the tool, the application will extract the following file to the same folder as CDFControl.exe:
- CDFControl.xml
(Configuration file)
How to Use CDFControl
See the user guide (CDFControl Menu > Help) for full usage instructions.
Note: Parse Trace consumes more memory than Parse Trace to File. Consider using Parse Trace to File when you parse large ETL files.
CDFControl AOT Features
AOT trace parsing
CDFControl v3.3 and later-supports parsing the always-on tracing (AOT) traces.
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To parse the AOT traces, do the same as common trace parsing.
AOT trace highlighting
You can highlight AOT traces for differentiation. Choose Tools and then Options. Select the highlighted AOT in the viewer check box to turn on/off AOT trace highlighting. AOT trace highlighting is enabled by default.
Security Permissions Required by CDFControl
Administrator privileges are required for all features to be functional.
Data Modified by CDFControl
- For dynamic TMF download, TMF files are temporarily stored in the user’s temp directory. These files are deleted automatically after use.
- If remote tracing is used, a few temporary files are generated.
For more information, see the Remote Tracing section in the user guide (CDFControl Menu > Help). - CDFControl captures traces. As a result, trace files will be present on the machine being traced.
- You can export trace template settings to XML and CTL files.
Uninstalling CDFControl
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To uninstall CDFControl and undo changes made to the system, delete all files generated in the same folder as CDFControl.exe.
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Contact Information
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Questions? Concerns? Send any feedback on this tool to CDFControl Feedback.