Let X and $($Xi$)$i in N be i$.$i$.$d random variables with distribution Xi $$ X that takes integer values. The random variable X takes integer values $$and represents the

number of descendants of any individual. We recursively define Zn $=(1,$ n $=0,$ X$1+$ X$2++$ XZn$1,$ n $1.$ The random variable Zn counts the number of individuals in generation n$.(+1$ point$)$ If $2=$ var$($X$),$ n $=$ E $[$Zn$]$ and $2$ n $=$ var$($Zn$)$ Using the identities $2$ n $=$ g $$ Zn $(1)+$ n $$ $2$ n $,(1)$ gZn $($s$)=$ gZn $($gx$($s$))1$ s $1.(2)2025-1$ Probability II $($a$)(+3$ point$)$ Prove the following recursive formula for $2$ n: $2$ n $=$ $2$ $2$ n$1+$ n$1$ $2.$

$($b$)(+1$ point$)$ Conclude that $2$ n $=$ n$1$ $2(1+$ $++$ n$1).$

and therefore $2$ n $=($ n $21+$ n $1$ $=1,$ n$2$ $=1.$

$$ Implement the following algorithm in Python or R and apply it to the situations described. To obtain points, you must submit a report with your

code and your analysis.