Exercise 1 Prove that the variance of a geometric random variable X with parameter p is: (16) 9-1/14 p² Hint Recall that Var[X] = E[X²] - E[X]2. Use Equation (8) to find E[X²]. Var[X] =
Exercise 1 Prove that the variance of a geometric random variable X with parameter p is: (16) 9-1/14 p² Hint Recall that Var[X] = E[X²] - E[X]2. Use Equation (8) to find E[X²]. Var[X] =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Exercise 1
![**Exercise 1**
Prove that the variance of a geometric random variable \( X \) with parameter \( p \) is:
\[ \text{Var}[X] = \frac{q}{p^2} \]
\[(16)\]
**Hint**
Recall that \(\text{Var}[X] = E[X^2] - E[X]^2\). Use Equation (8) to find \(E[X^2]\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F718b1378-40e4-4c32-83bc-211fc46d7de4%2F42816e3b-c1ea-4761-87d5-a7ef932e6707%2F1bab3qo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise 1**
Prove that the variance of a geometric random variable \( X \) with parameter \( p \) is:
\[ \text{Var}[X] = \frac{q}{p^2} \]
\[(16)\]
**Hint**
Recall that \(\text{Var}[X] = E[X^2] - E[X]^2\). Use Equation (8) to find \(E[X^2]\).
![**Theorem 2.1**
If \( X \) has a geometric distribution with parameter \( p \), then
\[ E[X] = \frac{1}{p} \]
(14)
**Proof**
Write
\[ E[X] = \sum_{x=1}^{\infty} x q^{x-1} p \]
\[ = \frac{p}{q} \sum_{x=0}^{\infty} x q^x \]
Using Equation (4) with \( a = q \)
\[ = \frac{p}{q} \left( \frac{q}{p^2} \right) = \frac{1}{p} \]
(15)
To give a numerical example, the average number of rolls of a pair of dice until the first roll of seven is 6.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F718b1378-40e4-4c32-83bc-211fc46d7de4%2F42816e3b-c1ea-4761-87d5-a7ef932e6707%2Fc759m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Theorem 2.1**
If \( X \) has a geometric distribution with parameter \( p \), then
\[ E[X] = \frac{1}{p} \]
(14)
**Proof**
Write
\[ E[X] = \sum_{x=1}^{\infty} x q^{x-1} p \]
\[ = \frac{p}{q} \sum_{x=0}^{\infty} x q^x \]
Using Equation (4) with \( a = q \)
\[ = \frac{p}{q} \left( \frac{q}{p^2} \right) = \frac{1}{p} \]
(15)
To give a numerical example, the average number of rolls of a pair of dice until the first roll of seven is 6.
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