Thread: Probability Bivariate RV Mean and Variance

Let X and Y be independent random variables with E(X)=E(Y)=5, var(X)=1 and var(Y)=(sigma^2)>1.

Put Z=aX + (1-a)Y, 0<=a<=1.

Find 1. the value of a that minimises var(A) and that minimum value.

2. the value of a that maximises var(Z) and that maximum value.

Help!

Originally Posted by maibs89

Let X and Y be independent random variables with E(X)=E(Y)=5, var(X)=1 and var(Y)=(sigma^2)>1.

Put Z=aX + (1-a)Y, 0<=a<=1.

Find 1. the value of a that minimises var(A) and that minimum value. Mr F says: What is the definition of the random variable A?

2. the value of a that maximises var(Z) and that maximum value.

Help!

Since X and Y are independent you can say that

Var(Z) = f(a) = a Var(X) + (1 - a) Var(Y).

Substitute for Var(X) and Var(Y) and

1. Minimise f over 0<=a<=1 ....... (assuming that A is a typo and you meant Z).

2. Maximise f over 0<=a<=1 .......

Note that .......

a is a constant.

I worked my way until Var(Z) = f(a) = a Var(X) + (1 - a) Var(Y).

Yes, I made a typo.

How do I find Var(Y)?

Originally Posted by maibs89

[snip]
How do I find Var(Y)?

You read the question: "....... var(Y)=(sigma^2)>1."