X be a geometric random variable, show that P [ X ≥ n ] = ( 1 − p ) n − 1

Explanation: P [ X ≥ x ] = ∑ x = n ∞ P [ X = x ] = ∑ x = n ∞ ( 1 − p ) x − 1 p = p 1 − p ∑ x = n ∞ ( 1 − p ) x = p 1 − p ( 1 − p ) n p Establishes the result.

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Roderick Bradley

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2022-08-16

X be a geometric random variable, show that

P [X \geq n] = (1 - p)^{n - 1}

Answer & Explanation

raffatoaq

Beginner2022-08-17Added 22 answers

Explanation:

P [X \geq x] = \sum_{x = n}^{\infty} P [X = x] = \sum_{x = n}^{\infty} (1 - p)^{x - 1} p = \frac{p}{1 - p} \sum_{x = n}^{\infty} (1 - p)^{x} = \frac{p}{1 - p} \frac{(1 - p)^{n}}{p}

Establishes the result.

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