作业帮求教一道统计学数学题.
来源：学生作业帮 编辑：作业帮 分类：数学作业 时间：2024/05/18 02:45:49
求教一道统计学数学题.
Suppose X and Y are random variables such that the atandard deviation of X is 2 and the standard deviation of Y is 5.Consider a random variable Z defined as Z=3X-2Y+5.问 a:Assume that X and Y are independtly distributed.Find the variance of Z
Suppose X and Y are random variables such that the atandard deviation of X is 2 and the standard deviation of Y is 5.Consider a random variable Z defined as Z=3X-2Y+5.问 a:Assume that X and Y are independtly distributed.Find the variance of Z
翻译：
X,Y是两个随机变量,X的标准差为2,Y的标准差为5.设随机变量Z=3X-2Y+5.假设X,Y独立,求Z的方差.
直接用方差的性质求
Var(Z)=3^2*Var(X)+(-2)^2*Var(Y)=9(σY)^2+4(σX)^2=9*(2^2)+4*(5^2)=136
再问： 谢谢你，还有第二问呢。 suppose that X and Y are not inpendent .Will you change your answer for a ? why
再答： 翻译：如果X，Y不独立，上一问的结果会不会改变？ 会改变。如果X，Y不独立，就要考虑X、Y之间的协方差，因为此时Var(Z)=3^2*Var(X)+(-2)^2*Var(Y)+2*3*(-2)*(X、Y之间的协方差)=136-12*(X、Y之间的协方差) 求协方差需要知道X、Y之间的关系，比如相关系数。所以这里没有办法求解。另外，也无法判断方差是否大于或者小于136，因为协方差可正可负。
再问： thx.anther Q is a fair coin is tossed twice.let X represent the number of heads obtained on the first toss and Y represent the total number of heads obtain in two tosses. a: derive the birvariate probability distribution for the variables X and Y. b: compute cov(X,Y) can you help me more?
再答： Yes, I can. But you'd better make it a new question, so that others can easily see it and can help you. I guess I don't have to translate, right? a: Tabular form for birvariate probability distribution: Y 0 1 2 X 0 1/4 1/4 0 1 0 1/4 1/4 P(X，Y)=1/4 (X=0, Y=0) 0 (X=1, Y=0) 1/4 (X=0, Y=1) 1/4 (X=1, Y=1) 0 (X=0, Y=2) 1/4 (X=1, Y=2) b: Use this formular for discrete random variables: cov(X,Y)=E(XY)-E(X)E(Y) E(X)=1/2*0+1/2*1=1/2 E(Y)=1/4*0+1/2*1+1/4*2=1 E(XY)=1/4*0*0+0*1*0+1/4*0*1+1/4*1*1+0*0*2+1/4*1*2=3/4 cov(X,Y)=E(XY)-E(X)E(Y)=3/4-1/2*1=1/4
X,Y是两个随机变量,X的标准差为2,Y的标准差为5.设随机变量Z=3X-2Y+5.假设X,Y独立,求Z的方差.
直接用方差的性质求
Var(Z)=3^2*Var(X)+(-2)^2*Var(Y)=9(σY)^2+4(σX)^2=9*(2^2)+4*(5^2)=136
再问： 谢谢你，还有第二问呢。 suppose that X and Y are not inpendent .Will you change your answer for a ? why
再答： 翻译：如果X，Y不独立，上一问的结果会不会改变？ 会改变。如果X，Y不独立，就要考虑X、Y之间的协方差，因为此时Var(Z)=3^2*Var(X)+(-2)^2*Var(Y)+2*3*(-2)*(X、Y之间的协方差)=136-12*(X、Y之间的协方差) 求协方差需要知道X、Y之间的关系，比如相关系数。所以这里没有办法求解。另外，也无法判断方差是否大于或者小于136，因为协方差可正可负。
再问： thx.anther Q is a fair coin is tossed twice.let X represent the number of heads obtained on the first toss and Y represent the total number of heads obtain in two tosses. a: derive the birvariate probability distribution for the variables X and Y. b: compute cov(X,Y) can you help me more?
再答： Yes, I can. But you'd better make it a new question, so that others can easily see it and can help you. I guess I don't have to translate, right? a: Tabular form for birvariate probability distribution: Y 0 1 2 X 0 1/4 1/4 0 1 0 1/4 1/4 P(X，Y)=1/4 (X=0, Y=0) 0 (X=1, Y=0) 1/4 (X=0, Y=1) 1/4 (X=1, Y=1) 0 (X=0, Y=2) 1/4 (X=1, Y=2) b: Use this formular for discrete random variables: cov(X,Y)=E(XY)-E(X)E(Y) E(X)=1/2*0+1/2*1=1/2 E(Y)=1/4*0+1/2*1+1/4*2=1 E(XY)=1/4*0*0+0*1*0+1/4*0*1+1/4*1*1+0*0*2+1/4*1*2=3/4 cov(X,Y)=E(XY)-E(X)E(Y)=3/4-1/2*1=1/4