statistics (expected value, variance)

\(\displaystyle \sum x\cdot p(x)\)

\(\displaystyle \sum x^{2}\cdot p(x)\)

You're only challenge is assigning probabilities. Let's see what you get.

n/a

dumbledore said:

Thanks for the reply

So a) E(x)= 840,000,000 (0.3) + 210,000,000 (0.7) = 399,000,000

I think your formula for variance is wrong, isnt is sum (x-E(x))^2 P(x)?

And when I use that I get b) variance= (840,000,000-399,000,000)^2 (0.3) + (210,000,000-399,000,000)^2 (0.7) which comes out to be a ridiculously large number

Click to expand...

What tkhunny meant was that all you need to do your computations are:

\(\displaystyle \displaystyle \sum_{i = 1}^n(X_i * p_i)\ and\ \sum_{i = 1}^n(X_i^2 * p_i).\)

\(\displaystyle Let\ \mu = E(X) = \displaystyle \left(\sum_{i=1}^nX_i * p_i\right).\)

\(\displaystyle Let\ \lambda = E(X^2) = \displaystyle \left(\sum_{i=1}^nX_i^2 * p_i\right).\)

\(\displaystyle \displaystyle E\{[X - E(X)]^2\} = E\{(X - \mu)^2\} = E\{X^2 - 2\mu X + \mu^2\} = E(X^2) + E(-2\mu X) + E(\mu^2) = \lambda + E(-2\mu X) + E(\mu^2) =\)

\(\displaystyle \displaystyle \lambda + \left\{\sum_{i=1}^n(- 2\mu ) * X_i * p_i\right\} + \left\{\sum_{i=1}^n\mu ^2 * p_i\right\} = \lambda + (- 2\mu ) \left\{\sum_{i = 1}^n(X_i * p_i)\right\} + \mu^2 \left\{\sum_{i=1}^np_i\right\} = \lambda + (- 2\mu * \mu) + (\mu^2 * 1) =\)

\(\displaystyle \lambda - 2\mu^2 + \mu^2 = \lambda - \mu^2 = E(X^2) - \{E(X)\}^2.\)

\(\displaystyle In\ short,\ E\{[X^2 - E(X)]^2\} = E(X^2) - \{E(X)\}^2.\)

dumbledore said:

Thanks for the reply

So a) E(x)= 840,000,000 (0.3) + 210,000,000 (0.7) = 399,000,000

I think your formula for variance is wrong, isnt is sum (x-E(x))^2 P(x)?

And when I use that I get b) variance= (840,000,000-399,000,000)^2 (0.3) + (210,000,000-399,000,000)^2 (0.7) which comes out to be a ridiculously large number

Click to expand...

1) Once you have the Moments, you are off and running.
2) Why on Earth would you actually type "840,000,000"? Why not scale a little? Maybe "840" would be sufficient. You can just remember that you are working in millions.
3) You seem to be considering only the case where the Tribunal goes well. What about the bad outcome?