【随机过程】课后作业 1-条件概率期望密度

深藏阁楼爱情的钟 2022-11-10 14:58 204阅读 0赞

[参考资料][Link 1]

# 1 相关链接 #

[【随机过程】课后作业2-随机足标和顺序统计量][2-]

# 2 习题 #

**1. 设N1，N2，N3独立，Ni是参数为i 的poisson分布，i=1，2，3.**  
（1）求P(N1+N2=3)  
（2）求E(N1|N2+N2=5)  
（3）求D(N1|N1+N2=5)  
解：  
（1）  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80MzkzNTY5Ng_size_16_color_FFFFFF_t_70_pic_center]  
带入n=3,计算求解P(N1+N2=3)=0.22404180765538775

（2）因为n有限，所以k=0、1、2…n  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80MzkzNTY5Ng_size_16_color_FFFFFF_t_70_pic_center 1]  
带入 λ 1 = 1 \\lambda\_\{1\}=1 λ1=1  λ 2 = 2 \\lambda\_\{2\}=2 λ2=2N1+N2=5,求得E(N1|N2+N2=5) =  5 3 \\frac\{5\}\{3\} 35  
（3）  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80MzkzNTY5Ng_size_16_color_FFFFFF_t_70_pic_center 2]

clear
    clc
    %期望
    EX =0;
    %平方的期望
    EX2 =0;
    for i=0:5
        p =  i*test(5,i,1,2);
        EX =EX+p;
    end
    for i=0:5
        p =  i*i*test(5,i,1,2);
        EX2 =EX2+p;
    end
    %方差
    DX =0
    DX = EX2-EX*EX
    function p =test(n,k,x1,x2)
        %factorial阶乘
        % x1对应题目的lambda1
        tempn = factorial(n);
        tempk = factorial(k);
        tempnk =factorial(n-k);
        %P分布函数
        p = (tempn*x1^k *x2^(n-k))/(tempk*tempnk*(x1+x2)^n)
    end

> 求解得到  
> Dx = 1.1111

**2. 设随机变量N服从参数为5的泊松分布，则求 Σ n = 0 ∞ \\Sigma\_\{n=0\}^\{\\infty\} Σn=0∞P(N>n)**  
解：由以下证明，得 Σ n = 0 ∞ \\Sigma\_\{n=0\}^\{\\infty\} Σn=0∞P(N>n) = EN  
又因为泊松分布的期望等于 λ \\lambda λ = 5  
所以EN = 5  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80MzkzNTY5Ng_size_16_color_FFFFFF_t_70_pic_center 3]

**3. 设随机变量X服从参数为2的卡方分布，分布函数为F(x)，则求 ∫ 0 ∞ 2 x ( 1 − F ( x ) ) d x \\displaystyle \\int^\{\\infty\}\_\{0\}\{2x(1-F(x))dx\} ∫0∞2x(1−F(x))dx**  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80MzkzNTY5Ng_size_16_color_FFFFFF_t_70_pic_center 4]

[Link 1]: https://wenku.baidu.com/view/d0f116fc770bf78a652954d9
[2-]: https://betterbench.blog.csdn.net/article/details/115003453
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