Question

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

Answer 1

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解：
Var(Z)=3^2*Var(X)+(-2)^2*Var(Y)=9(σY)^2+4(σX)^2=9*(2^2)+4*(5^2)=136

追问

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

追答

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。

追问

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?

追答

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

Answer 2

The answer is one,you are welcome

追问

can you help me anther Q