- #1

- 488

- 4

As many texts' discussion, we usually use a variable x for any value randomly picked. For a Bernoulli trials, i.e. each random variable x can either be successful or fail. If the probability of success if p and that of failure is q=1-p, then the expectation value of x would be

[tex]\langle x\rangle = x_s p + x_f(1-p)[/tex]

where [tex]x_s[/tex] is the value of success while [tex]x_f [/tex] is the value of failure.

In many texts, it takes [tex]x_s=1[/tex] and [tex]x_f=0[/tex]. Hence,

[tex]\langle x\rangle = x_s p + x_f(1-p) = p[/tex]

I wonder why and from what point shall we define success and failure as [tex]x_s=1[/tex] and [tex]x_f=0[/tex]? Why I can't say [tex]x_s=1[/tex] and [tex]x_f=-1[/tex] OR

[tex]x_s=0[/tex] and [tex]x_f=1[/tex]? But it we change the valus of [tex]x_s[/tex] and [tex]x_f[/tex], [tex]\langle x\rangle[/tex] will definitely be changed!?