Expectation and Standard Deviation(SD)

[helix999]

Hi

I have a question.

Let X1 & X2 be stochastic variables and X1<=X2, then can we say E[X1]<=E[X2] or SD[X1]<=SD[X2]? why or why not?

Looking forward to some reply

Thanks!

 

[lanedance]

what do you think?

and by X1 <=X2, does this mean every outcome of X1 is <= every outcome of X2?

 

[helix999]

yeah, the given condition is for every outcome.
I think E[x1]<=E[x2] but no idea abt std deviation. i don't know if i am correct.

 

[clamtrox]

I think E[x1]<=E[x2]

Could you prove it?

but no idea abt std deviation. i don't know if i am correct.

If you think that the relation does not always hold for standard deviation, could you perhaps find variables X and Y for which this is the case?

 

[lanedance]

its always a good place to start at the defintions (either discrete or continuous will work)

from the definitions, E[X1]<= E[X2] shoudl be obvious


for the standard deviation, building on clamtrox's argument, it might help to consider the dsicrete variable pdf:
P(X2= x2) = 1, if x2 = b
P(X2= x2) = 0, otherwise
whats the standard deviation of this "random" variable?

 

[statdad]

consider X1 and X2 with distributions
$$\begin{tabular}{l c c c c c}<br /> x & 10 & 20 & 30 & 40 & 50 \\<br /> p(x) & .80 & .1 & .07 & .02 & .01\\<br /> \end{tabular}$$

 

[helix999]

its always a good place to start at the defintions (either discrete or continuous will work)

from the definitions, E[X1]<= E[X2] shoudl be obvious


for the standard deviation, building on clamtrox's argument, it might help to consider the dsicrete variable pdf:
P(X2= x2) = 1, if x2 = b
P(X2= x2) = 0, otherwise
whats the standard deviation of this "random" variable?

zero.

 

[lanedance]

yep... just to show for a random variabe the SD, can be quite separate from the mean... hopefully you can thnk of a RV with all lesser outcomes, but non-zero SD

though i hope i haven't simplified too much, what exactly do you mean by a stochastic variable here...?

 

[helix999]

consider X1 and X2 with distributions
$$\begin{tabular}{l c c c c c}<br /> x & 10 & 20 & 30 & 40 & 50 \\<br /> p(x) & .80 & .1 & .07 & .02 & .01\\<br /> \end{tabular}$$

$$<br /> ok i got the examples when SD[x1]<=SD[x2]. But when SD[x1>=Sd[x2]?$$