Soit X une variable aleatoire de loi geometrique de parametre p∈]0.1[ calculer P({X≥4})


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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

	$P (X = k) = q^{k - 1} p = {(1 - p)}^{k - 1} p$ $P (X ⩾ 4) = 1 - P (X = 1) - P (X = 2) - P (X = 3)$ $P (X ⩾ 4) = 1 - {(1 - p)}^{0} p - (1 - p) p - {(1 - p)}^{2} p$ $P (X ⩾ 4) = 1 - p - (1 - p) p - {(1 - p)}^{2} p$ $P (X ⩾ 4) = (1 - p) (1 - p - (1 - p) p)$ $P (X ⩾ 4) = (1 - p) (1 - 2 p + p^{2})$ $P (X ⩾ 4) = (1 - p) {(1 - p)}^{2}$ $P (X ⩾ 4) = {(1 - p)}^{3}$
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