Probability theory question (mini max functions)

[kwy]

Homework Statement

EX=EY=5, VarX=1, VarY=sigma^2 >1
Z=aX+(1-a)Y, 0<=a<=1
find a that minimizes VarZ, and another a that maximize VarZ

Homework Equations

The Attempt at a Solution

Not even sure where to begin
*EX=5, VarX=1 thus EX^2 = 26
marginal px(x) =26/x^2 = 5/x but this finds me x not px(x)
*then I tried rewriting the equation a=(Z-Y)/(X-Y), but then I don't know how to utilise the mean and variance values.

Please help.

 

[Stephen Tashi]

Homework Statement

EX=EY=5, VarX=1, VarY=sigma^2 >1
Z=aX+(1-a)Y, 0<=a<=1

Did you omit the fact that X and Y are independent random variables?

find a that minimizes VarZ, and another a that maximize VarZ

Homework Equations

Let P and Q be independent random variables let k be a constant then
Let W = kP

Then
$$Var(W) = ?$$
$$Var(W + Q) = ?$$

From calculus:
To minimize $F(a)$ on the interval $0 \leq a \leq 1$ set $F'(a) = 0$ and solve for $a$. Compare the value of this $F(a)$ to the value of $F(0)$ and $F(1)$ to check for "endpoint extrema".

 

[kwy]

Yes, they are independent random variables. Can you please provide a bit more guidence? I just know Var(W) = kEX^2 -(kEX)^2? Am I totally off track?

 

[Stephen Tashi]

Don't your course materials prove theorems like

$var(kX) = k^2 Var(X)$ ?

 

[kwy]

Thanks for your patience, the course went so fast, I am still trying to revise all the notes between full time work.
so Var(W)= k^2Var(P)
does Var(W+Q)= VarW + VarQ + 2xCov(W,Q)??

 

[Ray Vickson]

Thanks for your patience, the course went so fast, I am still trying to revise all the notes between full time work.
so Var(W)= k^2Var(P)
does Var(W+Q)= VarW + VarQ + 2xCov(W,Q)??

Yes, to the last question. In fact, Var(a*X + b*Y) = a^2 * Var(X) + b^2*Var(Y) + 2*a*b*Cov(X,Y) for constants a and b.

RGV

 

[kwy]

Hi Ray, since the RVs X and Y are independent, I don't have to worry about the Cov(XY) right? Hence, the equation is now
VarZ = a^2VarX + (1-a)^2VarY (VarX = 1, VarY = sigma^2)
= a^2 + (1-a)^2*sigma^2 and where do the mean values come in?
How can I differentiate this?
Thanks in advance.

 

[Stephen Tashi]

Hi Ray, since the RVs X and Y are independent, I don't have to worry about the Cov(XY) right? Hence, the equation is now
VarZ = a^2VarX + (1-a)^2VarY (VarX = 1, VarY = sigma^2)
= a^2 + (1-a)^2*sigma^2 and where do the mean values come in?
How can I differentiate this?
Thanks in advance.

Don't let the fact that the variable is 'a' instead of 'x' confuse you. You could differentiate
$$F(x) = x^2 + (1-x)^2 s^2$$ with respect to $x$, couldn't you?

As to where the mean values come in, that's a good question! Is there a part 2 to this question?

 

[Ray Vickson]

Hi Ray, since the RVs X and Y are independent, I don't have to worry about the Cov(XY) right? Hence, the equation is now
VarZ = a^2VarX + (1-a)^2VarY (VarX = 1, VarY = sigma^2)
= a^2 + (1-a)^2*sigma^2 and where do the mean values come in?
How can I differentiate this?
Thanks in advance.

The means do not "come in"; that is the whole point of the formula!

As to the second question: you have a function of a. You should be able to differentiate it.

RGV

 

[kwy]

Thanks so much Ray and Stephen. It makes sense now.

 

[kwy]

Hi Guys, I think the means may be of use somewhere. There is no 2nd part to the question, the aim is to determine values of a that will i) minimise VarZ and ii) maximise VarZ. Following differentiation, a = sigma^2/(1+sigma^2). From this value and 2 random variables, how can I check the "endpoint extema"? Do I simply test with F(0,0) and F(1,1). Many thanks.

 

[Stephen Tashi]

The endpoint extrema are the values of Var(Z) = F(a) when a = 0 and when a = 1. (There's only one varaible 'a' here, so you don't need a notation suggesting that F has two variables.) There are 3 candidates for 'a': a = 0 , a = 1 or a = sigma^2 / (1 + sigma^2), each produces a different value for Var(Z).

 

[kwy]

Got it. Thanks heaps.

 

[Ray Vickson]

Hi Guys, I think the means may be of use somewhere. There is no 2nd part to the question, the aim is to determine values of a that will i) minimise VarZ and ii) maximise VarZ. Following differentiation, a = sigma^2/(1+sigma^2). From this value and 2 random variables, how can I check the "endpoint extema"? Do I simply test with F(0,0) and F(1,1). Many thanks.

Your function f(a) is a^2 + v*(1-a)^2, where v = sigma^2. The graph y = f(a) is an upward-opening parabola on the whole real a-line, so on 0 <= a <= 1, only three possibilities can occur: (i) f(a) is decreasing (i.e., non-increasing) as a function of a on the interval [0,1]; (ii) f(a) is increasing on [0,1]; or (iii) f(a) is decreasing on the left part of [0,1] and increasing on the right part. (Draw sketches to pin down your understanding.) Now you need to determine whether you are in case (i), (ii) or (iii), and then determine how to find the max and min of f(a) on the a-interval [0,1].

RGV

 

[kwy]

Many thanks.