1. Geometric Thoughts Starting with the probability mass function for the geometric distribution (“success" on the k +1 trial) f(k) = (1 – p)*p, k = 0, 1, 2, 3, .. %3D (a) Derive the cumulative mass function F(k) = 1 – – (1 – p)k+1 (b) Show that all of the probabilities do indeed add up to 100% in the following ways: i. Ef(k) k=0 ii. lim F(k) (c) Given |x| < 1, take the derivative of both sides of > 1 with respect to 1- x k=0 x.
1. Geometric Thoughts Starting with the probability mass function for the geometric distribution (“success" on the k +1 trial) f(k) = (1 – p)*p, k = 0, 1, 2, 3, .. %3D (a) Derive the cumulative mass function F(k) = 1 – – (1 – p)k+1 (b) Show that all of the probabilities do indeed add up to 100% in the following ways: i. Ef(k) k=0 ii. lim F(k) (c) Given |x| < 1, take the derivative of both sides of > 1 with respect to 1- x k=0 x.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![## Geometric Thoughts
Starting with the probability mass function for the geometric distribution (where "success" occurs on the \( k + 1 \) trial):
\[ f(k) = (1 - p)^k p, \quad k = 0, 1, 2, 3, \ldots \]
### Tasks:
(a) **Derive the cumulative mass function**
\[ F(k) = 1 - (1 - p)^{k+1} \]
(b) **Show that all of the probabilities do indeed add up to 100%** in the following ways:
i. Evaluate:
\[ \sum_{k=0}^{\infty} f(k) \]
ii. Evaluate:
\[ \lim_{k \to \infty} F(k) \]
(c) **Given \( |x| < 1 \), take the derivative of both sides** of:
\[ \sum_{k=0}^{\infty} x^k = \frac{1}{1-x} \]
with respect to \( x \).
(d) **Use the above result to derive the expected value** for the geometric distribution.
(e) **Using the complementary cumulative mass function**:
\[ P(X > k) = 1 - F(k) = (1 - p)^{k+1} \]
Prove the memoryless property.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F406074af-a425-40d0-923c-17adedf14fc2%2Fe5815d5c-46a1-4dca-9dad-0c6c69a6d33a%2F41t1ohm_processed.png&w=3840&q=75)
Transcribed Image Text:## Geometric Thoughts
Starting with the probability mass function for the geometric distribution (where "success" occurs on the \( k + 1 \) trial):
\[ f(k) = (1 - p)^k p, \quad k = 0, 1, 2, 3, \ldots \]
### Tasks:
(a) **Derive the cumulative mass function**
\[ F(k) = 1 - (1 - p)^{k+1} \]
(b) **Show that all of the probabilities do indeed add up to 100%** in the following ways:
i. Evaluate:
\[ \sum_{k=0}^{\infty} f(k) \]
ii. Evaluate:
\[ \lim_{k \to \infty} F(k) \]
(c) **Given \( |x| < 1 \), take the derivative of both sides** of:
\[ \sum_{k=0}^{\infty} x^k = \frac{1}{1-x} \]
with respect to \( x \).
(d) **Use the above result to derive the expected value** for the geometric distribution.
(e) **Using the complementary cumulative mass function**:
\[ P(X > k) = 1 - F(k) = (1 - p)^{k+1} \]
Prove the memoryless property.
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