Recap from Probability

Expectation formulas:

• Discrete random variable $X$: $E[X]=\sum_ia_iP(X+a_i)$
  • $E[g(X)]=\sum_ig(a_i)P(X=a_i)$
• Continuous random variable $X$: $E[X]=\int^{+\infin}_{-\infin}xf(x)dx$
  • $E[g(X)]=\int^{+\infin}_{-\infin}g(x)f(x)dx$

Note that: $E[rX+s]=rE[X]+s$

Variance formula: $Var(X)=E[(X-E[X])^2]=E[X^2]-(E[X])^2$

Noe that $Var(rX+s)=r^2Var(X)$

Standard deviation formula: $SD(X)=\sqrt{Var(X)}$

If we have 2 random variables $X$ and $Y$:

• If they are discrete:
  • $E[g(X,Y)]=\sum_i\sum_jg(a_i,b_j)P(X=a_i,Y=b_j)$
• If they are continuous:
  • $E[g(X,Y)]=\int^{+\infin}{-\infin}\int^{+\infin}{-\infin} g(x,y)f(x,y)dxdy$

$E[rX+sY+t]=rE[X]+sE[Y]+t$

If $X$ and $Y$ are independent, then $E[XY]=E[X]E[Y]$

$Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)$.

If $X,Y$ are uncorrelated (or independent), then $Var(X+Y)=Var(X)+Var(Y)$.

Finally, remember that, give some $X\sim N(\mu,\sigma^2)$, then $Z=\frac{X-\mu}{\sigma}\sim N(0,1)$.

Unbiased estimators

Suppose you have a bunch of $12$ radioactive atoms. We know, because of physics, that the time until an isotope decays is exponentially distributed, with a parameter $\lambda$ depending on the half-life of the isotope. We measure for each of the atoms when it decays: