Assume that Bernoulli trials are to be performed until a first success occurs. Denote by X the number of trials thus performed and by p the...

Assume that Bernoulli trials are to be performed until a first success occurs. Denote by X the number of trials thus performed and by p the probability of a success occurring on any given trial.

Next, X more Bernoulli trials are performed.

Finally, let Y denote the number of observed successes in this second group of trials.

Show that

E(Y)=1   and   Var(Y)=2(1−p).