properties of mean and variance

[maxleng]

do we need to know these? i came across a Q in NEAP 2006 exam 1

evaluate E(2x + 1)

oh and if anyone knows can you please explain how to do it, thnx

[Collin Li]

E(aX+b) = a*E(X) + b

It makes sense because if you think about this table:

Code: [Select]

x      0    1     2
Pr(x)  0.5  0.25  0.25

E(X) = 0 + 1(0.25) + 2(0.25) = 0.75
E(2X+1) = 1(0.5) + 3(0.25) + 5(0.25) = 2.5

Think about 2x+1, you'll have the values: 1, 3 and 5. Your mean will be multiplied by 2, and 1 will be added onto it. The +1 makes sense because all your values get shifted by 1, the multiplication makes sense because all your values got multiplied by that number.

There's also: Var(aX+b) = a^2Var(X)

Note that b has no effect. Variance (the spread of the system) is not affected by a translation of the elements.

[bilgia]

E(2X)+E(1)
2E(X) + 1

[maxleng]

ahh i see, thanks guys

another Question:

how do i find the  range of a composite function, i know the domain is the domain of the inside function but im lost for the range?

eg,
f(x) = 3sin(2x); 0<x<pi
g(x) = 1 - x^2; R

Range of : g(f(x)) = 1 - (3sin(2x))^2 ?

[Collin Li]

There's no rule, I think:

You have to think about the interval of the domain, and see what values ranges between that. Consider the endpoints, and turning points, and take the minimum and maximum from that interval, and that is your range.

[Collin Li]

So for 1 - 9sin^2(2x), for x is an element of (0, pi):

Note that 0, pi is a full oscillation for sin(2x) (sub in 0 gives you 0, sub in pi gives you 2pi).

A full oscillation ranges from -1 to 1, but remember that the function is sin^2(2x). Since it's squared, the range of sin^2(2x) is described as 0 to 1.

Now, take 1 - 9(0) and 1 - 9(1) [subbing sin^2(2x) into the function], and that is your range: (-8, 1)