Need clarifications on University Statistics course notes

(This comes from a former student of mine now in her second year at Singapore Management University (SMU)-I received this e-mail earlier on 9 September 2013):

Hi Mr Koh!

XXX  here. How are you? School has started and stats has come back to haunt. Have a couple of questions, hope you won't mind answering them! 

For starters, what is this?  

Attached are my lesson slides for context if needed. 

(Stats 101 Powerpoint slides)

Thanks so much and hope to hear from you soon!

Warm regards,

Former Student XXX

Hi ,                     

So the new semester has started; nice ppt slides you got there btw. Let me furnish examples to help you understand things better:                    

(i) E(Y) when Y is a discrete variable

Let's say we have Y=X^2 and the probability distribution function (pdf) of X is as follows:  

 X              0               1             2          3          4 

P(X=x)         0            1/6         1/3        1/12       5/12

Then the probability distribution function (pdf) of Y would be:   

Y               0                 1           4           9          16 

P(Y=y)           0             1/6       1/3      1/12     5/12

In that regard, E(Y) = 0(0) +(1)(1/6) + 4(1/3) + 9 (1/12) + 16 (5/12) = 8.9167 (shown)                 

(ii) E(Y) when Y is a continuous random variable

Let's say X is defined by the following probability distribution function:

f(x) =  2x + 1        0 ≦ x ≦ 1,         0    otherwise  

and Y is related to X such that   Y=X^2, 

then E(Y) = E(X^2) is obtained by evaluating the integral  ∫ (x^2) * (2x+1) dx  from x=0 to x=1. 

Hope this clarifies. Welcome back to the realm of Statistics.  

Best Regards,

Mr Koh