CT1 Ch15 Q15.2

[saksham]

I am not at all able to understand q 15.3 pg 6. Anyone please explain

 

[vgarg]

We have \(i_k, \quad k\in \{0,1,\ldots n\}\) i.i.d rvs with

\( p(i_k = a) =\begin{cases} \frac{1}{3} & \text{ for } a\in \{0.02,0.04,0.06\} \\0 \text{otherwise} \end{cases}\)

and \(S_n = \prod_{k=1}^{n} (1+i_k)\)

We have been asked to find \( \Pr(S_n = 1.02\times 1.04^{n-1})\)

Since \(S_n\) can take \(3^n\) values with each value being equally likely and there are \(\binom{n}{1}\) possibilities where for one of the year interest rate is \(0.2\) and for \(n-1\) years it is \(0.04\)

therefore \(\Pr(S_n = 1.02\times 1.04^{n-1}) = \dfrac{n}{3^n}\)