blocks|key|2784439|text|混合|type|unstyled|depth|inlineStyleRanges|offset|length|style|BOLD|entityRanges|data|2784440|两个随机变量的mixture均值概率为p使用分布1，概率为1-p使用分布2。|2784441|根据你的图表，你似乎是在混合发行版，而不是添加(卷积)它们。精确的结果对混合概率非常重要。例如，我选择了a+=+0.25、b+=+0.35和c+=+1-a-b。|CODE|2784442|对于混合物，probability+density+function+(PDF)是分析性可用的：|2784443|pdfMix+=@(x)+a.*pdf(pd1,x)+%2B+b.*pdf(pd2,x)+%2B+c.*pdf(pd3,x)。|2784444|%25+MATLAB+R2018b
pd1+=+makedist('Uniform',2,6);
pd2+=+makedist('Uniform',2,4);
pd3+=+makedist('Uniform',5,6);
a+=+0.25;
b+=+0.35;
c+=+1+-+a+-+b;++++%25+a+%2B+b+%2B+c+=+1

pdfMix+=@(x)+a.*pdf(pd1,x)+%2B+b.*pdf(pd2,x)+%2B+c.*pdf(pd3,x);

Xrng+=+0:.01:8;
plot(Xrng,pdfMix(Xrng))
xlabel('X')
ylabel('Probability+Density+Function')|code-block|syntax|javascript|2784445|​|2784446|📷|atomic|2784447|2784448|由于混合的发行版是统一的，您还可以使用stairs()命令：stairs(Xrng,pdfMix(Xrng))。|2784449|我们可以通过确保总面积为1来验证这是有效的PDF。|2784450|integral(pdfMix,0,9)|2784451|2784452|+ans+=+1.0000|blockquote|2784453|2784454|2784455|2784456|卷积:添加随机变量|2784457|将随机变量加在一起会产生不同的结果。同样，这可以通过经验很容易地完成。这在分析上是可能的。例如，将两个均匀的(0，1)分布卷积得到三角形(0，1，2)分布。随机变量的convolution只是一种奇特的方式，表示我们将它们相加，如果你对分析结果感兴趣，有一种方法可以使用积分获得最终的。|2784458|N+=+80000;++++++++++++++++++%25+Number+of+samples
X1+=+random(pd1,N,1);+++++++%25+Generate+samples+++++
X2+=+random(pd2,N,1);
X3+=+random(pd3,N,1);

X+=+X1+%2B+X2+%2B+X3;+++++++++++%25+Convolution++++++|2784459|请注意x轴(Xrng+=+0:.01:16;)的比例更改。|2784460|2784461|2784462|2784463|为了获得这一点，我使用random()从每个分布中生成80k个样本，然后将它们相加，以获得所需卷积的80k个样本。注意，当我使用histogram()时，我使用了'Normalization',+'pdf'选项。|2784464|Xrng+=+0:.01:16;
figure,+hold+on,+box+on
p(1)+=+plot(Xrng,pdf(pd1,Xrng),'DisplayName','X1+\sim+U(2,6)')
p(2)+=+plot(Xrng,pdf(pd2,Xrng),'DisplayName','X2+\sim+U(2,4)')
p(3)+=+plot(Xrng,pdf(pd3,Xrng),'DisplayName','X3+\sim+U(5,6)')
h+=+histogram(X,'Normalization','pdf','DisplayName','X+=+X1+%2B+X2+%2B+X3')

%25+Cosmetics
legend('show','Location','northeast')
for+k+=+1:3
++++p(k).LineWidth+=+2.0;
end
title('X+=+X1+%2B+X2+%2B+X3+(50k+samples)')
xlabel('X')
ylabel('Probability+Density+Function+(PDF)')|2784465|您可以使用fitdist()和Kernel+distribution+object获得PDF的估计值，然后在生成的内核分布对象上调用pdf()命令。|2784466|pd_kernel+=+fitdist(X,'Kernel')

figure,+hold+on,+box+on
h+=+histogram(X,'Normalization','pdf','DisplayName','X+=+X1+%2B+X2+%2B+X3')
pk+=+plot(Xrng,pdf(pd_kernel,Xrng),'b-')+++++++++++%25+Notice+use+of+pdf+command
legend('Empirical','Kernel+Distribution','Location','northwest')|2784467|如果这样做，您将注意到生成的内核是无界的，但是您可以纠正这一点，因为您知道使用truncate()的界限。您也可以使用ksdensity()函数，不过由于您可以直接访问all+the+functions，因此概率分布对象方法可能对用户更友好。您应该知道内核是近似值(您可以在内核图中清楚地看到这一点)。在这种情况下，卷积3个均匀分布的集成并不是太差，所以如果需要PDF，解析地找到PDF可能是首选。否则，经验方法(特别是对于生成)可能就足够了，尽管这取决于您的应用程序。|2784468|pdt_kernel+=+truncate(pd_kernel,9,16)|2784469|2784470|2784471|2784472|从混合和卷积中生成样本是一个不同的问题(但可以管理)。|2784473|entityMap|0|LINK|mutability|MUTABLE|url|https://en.wikipedia.org/wiki/Mixture_distribution|1|https://en.wikipedia.org/wiki/Probability_density_function|2|IMAGE|IMMUTABLE|imageUrl|https://ask.qcloudimg.com/http-save/yehe-900000/79e21fd98558d738eaa22d2e5497463d.jpeg|imageAlt|3|https://www.mathworks.com/help/matlab/ref/stairs.html|4|https://en.wikipedia.org/wiki/Convolution_of_probability_distributions|5|https://ask.qcloudimg.com/http-save/yehe-900000/12b680bd2db06869f592d6ce625b607e.jpeg|6|https://www.mathworks.com/help/stats/prob.normaldistribution.random.html|7|https://www.mathworks.com/help/matlab/ref/matlab.graphics.chart.primitive.histogram.html|8|https://www.mathworks.com/help/stats/fitdist.html|9|https://www.mathworks.com/help/stats/prob.kerneldistribution.html|10|https://www.mathworks.com/help/stats/prob.normaldistribution.pdf.html|11|https://www.mathworks.com/help/sldo/ref/prob.normaldistribution.truncate.html|12|https://www.mathworks.com/help/stats/ksdensity.html|13|https://www.mathworks.com/help/stats/working-with-probability-distributions.html|14|https://ask.qcloudimg.com/http-save/yehe-900000/64042d0f69ddac205c825fcf09de3db6.jpeg^0|0|2|0|7|7|0|0|10|4|1G|8|1P|8|1Y|9|0|6|Y|1|0|0|1M|0|0|0|0|1|2|0|0|J|8|U|P|J|8|3|0|0|0|K|0|0|0|0|0|0|0|9|0|2B|B|4|0|0|6|G|0|0|0|1|5|0|0|B|8|1S|B|29|M|B|8|6|1S|B|7|0|0|5|9|1U|5|5|9|8|F|Q|9|1U|5|A|0|0|13|A|1N|B|13|A|B|1N|B|C|2C|H|D|0|0|0|0|1|E|0|0|0^^$0|@$1|2|3|4|5|6|7|38|8|@$9|39|A|3A|B|C]]|D|@]|E|$]]|$1|F|3|G|5|6|7|3B|8|@]|D|@$9|3C|A|3D|1|3E]]|E|$]]|$1|H|3|I|5|6|7|3F|8|@$9|3G|A|3H|B|C]|$9|3I|A|3J|B|J]|$9|3K|A|3L|B|J]|$9|3M|A|3N|B|J]]|D|@]|E|$]]|$1|K|3|L|5|6|7|3O|8|@]|D|@$9|3P|A|3Q|1|3R]]|E|$]]|$1|M|3|N|5|6|7|3S|8|@$9|3T|A|3U|B|J]]|D|@]|E|$]]|$1|O|3|P|5|Q|7|3V|8|@]|D|@]|E|$R|S]]|$1|T|3|U|5|6|7|3W|8|@]|D|@]|E|$]]|$1|V|3|W|5|X|7|3X|8|@]|D|@$9|3Y|A|3Z|1|40]]|E|$]]|$1|Y|3|U|5|6|7|41|8|@]|D|@]|E|$]]|$1|Z|3|10|5|6|7|42|8|@$9|43|A|44|B|J]|$9|45|A|46|B|J]]|D|@$9|47|A|48|1|49]]|E|$]]|$1|11|3|12|5|6|7|4A|8|@]|D|@]|E|$]]|$1|13|3|14|5|6|7|4B|8|@$9|4C|A|4D|B|J]]|D|@]|E|$]]|$1|15|3|-4|5|6|7|4E|8|@]|D|@]|E|$]]|$1|16|3|17|5|18|7|4F|8|@]|D|@]|E|$]]|$1|19|3|-4|5|6|7|4G|8|@]|D|@]|E|$]]|$1|1A|3|-4|5|6|7|4H|8|@]|D|@]|E|$]]|$1|1B|3|-4|5|6|7|4I|8|@]|D|@]|E|$]]|$1|1C|3|1D|5|6|7|4J|8|@$9|4K|A|4L|B|C]]|D|@]|E|$]]|$1|1E|3|1F|5|6|7|4M|8|@]|D|@$9|4N|A|4O|1|4P]]|E|$]]|$1|1G|3|1H|5|Q|7|4Q|8|@]|D|@]|E|$R|S]]|$1|1I|3|1J|5|6|7|4R|8|@$9|4S|A|4T|B|J]]|D|@]|E|$]]|$1|1K|3|U|5|6|7|4U|8|@]|D|@]|E|$]]|$1|1L|3|W|5|X|7|4V|8|@]|D|@$9|4W|A|4X|1|4Y]]|E|$]]|$1|1M|3|U|5|6|7|4Z|8|@]|D|@]|E|$]]|$1|1N|3|1O|5|6|7|50|8|@$9|51|A|52|B|J]|$9|53|A|54|B|J]|$9|55|A|56|B|J]]|D|@$9|57|A|58|1|59]|$9|5A|A|5B|1|5C]]|E|$]]|$1|1P|3|1Q|5|Q|7|5D|8|@]|D|@]|E|$R|S]]|$1|1R|3|1S|5|6|7|5E|8|@$9|5F|A|5G|B|J]|$9|5H|A|5I|B|J]]|D|@$9|5J|A|5K|1|5L]|$9|5M|A|5N|1|5O]|$9|5P|A|5Q|1|5R]]|E|$]]|$1|1T|3|1U|5|Q|7|5S|8|@]|D|@]|E|$R|S]]|$1|1V|3|1W|5|6|7|5T|8|@$9|5U|A|5V|B|J]|$9|5W|A|5X|B|J]]|D|@$9|5Y|A|5Z|1|60]|$9|61|A|62|1|63]|$9|64|A|65|1|66]]|E|$]]|$1|1X|3|1Y|5|Q|7|67|8|@]|D|@]|E|$R|S]]|$1|1Z|3|U|5|6|7|68|8|@]|D|@]|E|$]]|$1|20|3|W|5|X|7|69|8|@]|D|@$9|6A|A|6B|1|6C]]|E|$]]|$1|21|3|U|5|6|7|6D|8|@]|D|@]|E|$]]|$1|22|3|23|5|6|7|6E|8|@]|D|@]|E|$]]|$1|24|3|-4|5|6|7|6F|8|@]|D|@]|E|$]]]|25|$26|$5|27|28|29|E|$2A|2B]]|2C|$5|27|28|29|E|$2A|2D]]|2E|$5|2F|28|2G|E|$2H|2I|2J|-4]]|2K|$5|27|28|29|E|$2A|2L]]|2M|$5|27|28|29|E|$2A|2N]]|2O|$5|2F|28|2G|E|$2H|2P|2J|-4]]|2Q|$5|27|28|29|E|$2A|2R]]|2S|$5|27|28|29|E|$2A|2T]]|2U|$5|27|28|29|E|$2A|2V]]|2W|$5|27|28|29|E|$2A|2X]]|2Y|$5|27|28|29|E|$2A|2Z]]|30|$5|27|28|29|E|$2A|31]]|32|$5|27|28|29|E|$2A|33]]|34|$5|27|28|29|E|$2A|35]]|36|$5|2F|28|2G|E|$2H|37|2J|-4]]]]

<h1>Mixture</h1>
A <a href="https://en.wikipedia.org/wiki/Mixture_distribution" rel="nofollow noreferrer">mixture</a> of two random variables means with probability p use Distribution 1, and with probability 1-p use Distribution 2.
Based on your graph, it appears you are mixing the distributions rather than adding (convolving) them. The precise results matter very much upon the mixing probabilities. As an example, I've chosen <code>a = 0.25</code>, <code>b = 0.35</code>, and <code>c = 1-a-b</code>.
For a mixture, the <a href="https://en.wikipedia.org/wiki/Probability_density_function" rel="nofollow noreferrer">probability density function (PDF)</a> is analytically available: 
<code>pdfMix =@(x) a.*pdf(pd1,x) + b.*pdf(pd2,x) + c.*pdf(pd3,x)</code>.
<pre><code>% MATLAB R2018b
pd1 = makedist('Uniform',2,6);
pd2 = makedist('Uniform',2,4);
pd3 = makedist('Uniform',5,6);
a = 0.25;
b = 0.35;
c = 1 - a - b; % a + b + c = 1

pdfMix =@(x) a.*pdf(pd1,x) + b.*pdf(pd2,x) + c.*pdf(pd3,x);

Xrng = 0:.01:8;
plot(Xrng,pdfMix(Xrng))
xlabel('X')
ylabel('Probability Density Function')
</code></pre>
<a href="https://i.stack.imgur.com/qC2Fk.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/qC2Fk.jpg" alt="Mixture probability density function." /></a>
Since the distributions being mixed are uniform you could also use the <a href="https://www.mathworks.com/help/matlab/ref/stairs.html" rel="nofollow noreferrer"><code>stairs()</code></a> command: <code>stairs(Xrng,pdfMix(Xrng))</code>.
We can verify this is a valid PDF by ensuring the total area is 1. 
<code>integral(pdfMix,0,9)</code>
<blockquote>
ans = 1.0000
</blockquote>
<hr />
<h1>Convolution: Adding Random Variables</h1>
Adding the random variables together yields a different result. Again, this can be done empirically easily. It is possible to this analytically. For example, convolving two Uniform(0,1) distributions yields a Triangular(0,1,2) distribution. The <a href="https://en.wikipedia.org/wiki/Convolution_of_probability_distributions" rel="nofollow noreferrer">convolution</a> of random variables is just a fancy way of saying we add them up and there is a way to obtain the resulting PDF using integration if you're interested in analytical results.
<pre><code>N = 80000; % Number of samples
X1 = random(pd1,N,1); % Generate samples 
X2 = random(pd2,N,1);
X3 = random(pd3,N,1);

X = X1 + X2 + X3; % Convolution 
</code></pre>
Notice the change of scale for the x-axis (<code>Xrng = 0:.01:16;</code>).
<a href="https://i.stack.imgur.com/kKMHe.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/kKMHe.jpg" alt="Resulting histogram from convolution (adding RVs)" /></a>
To obtain this, I generated 80k samples from each distribution with <a href="https://www.mathworks.com/help/stats/prob.normaldistribution.random.html" rel="nofollow noreferrer"><code>random()</code></a> then added them up to obtain 80k samples of the desired convolution. Notice when I used <a href="https://www.mathworks.com/help/matlab/ref/matlab.graphics.chart.primitive.histogram.html" rel="nofollow noreferrer"><code>histogram()</code></a> I used the <code>'Normalization', 'pdf'</code> option.
<pre><code>Xrng = 0:.01:16;
figure, hold on, box on
p(1) = plot(Xrng,pdf(pd1,Xrng),'DisplayName','X1 \sim U(2,6)')
p(2) = plot(Xrng,pdf(pd2,Xrng),'DisplayName','X2 \sim U(2,4)')
p(3) = plot(Xrng,pdf(pd3,Xrng),'DisplayName','X3 \sim U(5,6)')
h = histogram(X,'Normalization','pdf','DisplayName','X = X1 + X2 + X3')

% Cosmetics
legend('show','Location','northeast')
for k = 1:3
 p(k).LineWidth = 2.0;
end
title('X = X1 + X2 + X3 (50k samples)')
xlabel('X')
ylabel('Probability Density Function (PDF)')
</code></pre>
You can obtain an estimate of the PDF using the <a href="https://www.mathworks.com/help/stats/fitdist.html" rel="nofollow noreferrer"><code>fitdist()</code></a> and the <a href="https://www.mathworks.com/help/stats/prob.kerneldistribution.html" rel="nofollow noreferrer">Kernel distribution object</a> then calling the <a href="https://www.mathworks.com/help/stats/prob.normaldistribution.pdf.html" rel="nofollow noreferrer"><code>pdf()</code></a> command on the resulting Kernel distribution object.
<pre><code>pd_kernel = fitdist(X,'Kernel')

figure, hold on, box on
h = histogram(X,'Normalization','pdf','DisplayName','X = X1 + X2 + X3')
pk = plot(Xrng,pdf(pd_kernel,Xrng),'b-') % Notice use of pdf command
legend('Empirical','Kernel Distribution','Location','northwest')
</code></pre>
If you do this, you'll notice the resulting kernel is unbounded but you can correct this since you know the bounds using <a href="https://www.mathworks.com/help/sldo/ref/prob.normaldistribution.truncate.html" rel="nofollow noreferrer"><code>truncate()</code></a>. You could also use the <a href="https://www.mathworks.com/help/stats/ksdensity.html" rel="nofollow noreferrer"><code>ksdensity()</code></a> function, though the probability distribution object approach is probably more user friendly due to <a href="https://www.mathworks.com/help/stats/working-with-probability-distributions.html" rel="nofollow noreferrer">all the functions</a> you have direct access to. You should be aware that the kernel is an approximation (you can see that clearly in the kernel plot). In this case, the integration to convolve 3 uniform distributions isn't too bad, so finding the PDF analytically is probably the preferred choice if the PDF is desired. Otherwise, empirical approaches (especially for generation), are probably sufficient though this depends on your application.
<pre><code>pdt_kernel = truncate(pd_kernel,9,16)
</code></pre>
<h2><a href="https://i.stack.imgur.com/JOyQA.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/JOyQA.jpg" alt="Kernel density" /></a></h2>
Generating samples from mixtures and convolutions is a different issue (but manageable).

I need to reproduce the normalized density p(x) below, but the code given does not generate a normalized PDF.

<a href="https://i.stack.imgur.com/NlE28.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/NlE28.png" alt="A plot of p of x"></a>

<pre><code>clc, clear
% Create three distribution objects with different parameters
pd1 = makedist('Uniform','lower',2,'upper',6);
pd2 = makedist('Uniform','lower',2,'upper',4);
pd3 = makedist('Uniform','lower',5,'upper',6);
% Compute the pdfs
x = -1:.01:9;
pdf1 = pdf(pd1,x); 
pdf2 = pdf(pd2,x); 
pdf3 = pdf(pd3,x); 
% Sum of uniforms
pdf = (pdf1 + pdf2 + pdf3);
% Plot the pdfs
figure;
stairs(x,pdf,'r','LineWidth',2);
</code></pre>

If I calculate the normalized mixture PDF by simply scaling them by their total sum, I have different normalized probability comparing with the original figure above.

<pre><code>pdf = pdf/sum(pdf);
</code></pre>

Plot normalized uniform mixture

Kernel

我需要重现下面的归一化密度p(x)，但是给出的代码不会生成归一化的PDF。​​clc, clear% Create three distribution objects with different parameterspd1 = makedist('Uniform','lower',2,'upper',6);pd2 ...

问绘制归一化均匀混合图
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