Question

The random variable X has a geometric distribution with parameter p. (a) Show that $P(X \le n) = 1 - (1 - p)^n$, $n \in \mathbb{Z}^+$. [3 marks] (b) Deduce an expression for $P(m < X \le n)$, $m, n \in \mathbb{Z}^+$ and $m < n$. [1 mark] (c) Given that $p = 0.2$, find the least value of n for which $P(1 < X \le n) > 0.5$, $n \in \mathbb{Z}^+$. [2 marks]

          The random variable X has a geometric distribution with parameter p.
(a) Show that $P(X \le n) = 1 - (1 - p)^n$, $n \in \mathbb{Z}^+$.
[3 marks]
(b) Deduce an expression for $P(m < X \le n)$, $m, n \in \mathbb{Z}^+$ and $m < n$.
[1 mark]
(c) Given that $p = 0.2$, find the least value of n for which
$P(1 < X \le n) > 0.5$, $n \in \mathbb{Z}^+$.
[2 marks]

The random variable X has a geometric distribution with parameter p.
(a) Show that P(X ≤ n) = 1 - (1 - p)^n, n ∈ℤ^+.
[3 marks]
(b) Deduce an expression for P(m < X ≤ n), m, n ∈ℤ^+ and m < n.
[1 mark]
(c) Given that p = 0.2, find the least value of n for which
P(1 < X ≤ n) > 0.5, n ∈ℤ^+.
[2 marks]

Added by Sara F.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Robin Corrigan

02/09/2024

Step 1

This can be done by using the definition of the geometric distribution, which states that P(X = k) = p(1 - p)^(k-1) for k = 1, 2, 3, ... Show more…

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Hi, I will help you with this. Let's start by correcting the errors in the text provided: Original: (a) Show that P(x<=n)=1-(1-p)^(n),ninZ^(+). (b) Deduce an expression for p=0.2n The random variable X has a geometric distribution with parameter p (a) Show that P(Xn)=1-(l-p)", neZ*. [3 marks] (b) Deduce an expression for P(m<Xn), m, ne Z* and m<n. [1 mark] (c) Given that p=0.2. find the least value of n for which P(1<Xn)>0.5,neZ+ [2 marks] Corrected: (a) Show that P(x<=n) = 1 - (1 - p)^n, n ∈ Z^+. (b) Deduce an expression for P(X < n) = 1 - (1 - p)^n, n ∈ Z^*. [3 marks] (b) Deduce an expression for P(m < X < n), m, n ∈ Z* and m < n. [1 mark] (c) Given that p = 0.2, find the least value of n for which P(1 < X < n) > 0.5, n ∈ Z^+ [2 marks] I hope this helps! Let me know if you need further clarification.