Soit-X-une-variable-aleatoire-de-loi-geometrique-de-parametre-p-0-1-calculer-P-X-4-

Question Number 145136 by ArielVyny last updated on 02/Jul/21

S o i t X u n e v a r i a b l e a l e a t o i r e d e l o i

g e o m e t r i q u e d e p a r a m e t r e p \in] 0.1 [

c a l c u l e r P ({X ⩾ 4})

Answered by Olaf_Thorendsen last updated on 03/Jul/21

P (X = k) = q^{k - 1} p = {(1 - p)}^{k - 1} p

P (X ⩾ 4) = 1 - P (X = 1) - P (X = 2) - P (X = 3)

P (X ⩾ 4) = 1 - {(1 - p)}^{0} p - (1 - p) p - {(1 - p)}^{2} p

P (X ⩾ 4) = 1 - p - (1 - p) p - {(1 - p)}^{2} p

P (X ⩾ 4) = (1 - p) (1 - p - (1 - p) p)

P (X ⩾ 4) = (1 - p) (1 - 2 p + p^{2})

P (X ⩾ 4) = (1 - p) {(1 - p)}^{2}

P (X ⩾ 4) = {(1 - p)}^{3}