(ii) Is X a geometric random variable? If so, then specify the success probability

q of X. Explain.

To determine whether \( X \) is a geometric random variable and, if so, to specify the success probability \( q \) of \( X \), we need to analyze the distribution of \( X \) in the context of the problem.

### **Recap of the Problem:**
- \( U \sim U[0, 1] \) is a uniform random variable on the interval \([0, 1]\).
- The random variable \( X \) has a conditional distribution given \( U = p \), where \( X \) given \( U = p \) follows a geometric distribution with success probability \( p \):
  \[
  P(X = n \mid U = p) = p(1-p)^{n-1}, \quad n = 1, 2, 3, \dots
  \]
  
### **Key Question: Is \( X \) a Geometric Random Variable?**

A geometric random variable \( Y \) with success probability \( q \) has the PMF:
\[
P(Y = n) = q(1-q)^{n-1}, \quad n = 1, 2, 3, \dots
\]

To determine whether \( X \) is geometric, we need to check whether the marginal distribution \( P(X = n) \) has the same form as the PMF above for some success probability \( q \).

### **Marginal Distribution of \( X \)**

From the previous part, we computed \( P(X = 1) \) and \( P(X = 2) \) as:
- \( P(X = 1) = \frac{1}{2} \)
- \( P(X = 2) = \frac{1}{6} \)

Now, let's generalize the process for any \( n \):
\[
P(X = n) = \int_{0}^{1} P(X = n \mid U = p) f_U(p) \, dp
\]
\[
P(X = n) = \int_{0}^{1} p(1-p)^{n-1} \, dp
\]

### **Form of the Marginal Distribution:**

The probability \( P(X = n) \) is given by the integral:
\[
P(X = n) = \int_{0}^{1} p(1-p)^{n-1} \, dp
\]
This does not simplify to the geometric form \( q(1-q)^{n-1} \) for some fixed \( q \). Instead, it involves an integral that depends on \( n \).

### **Conclusion:**

- **\( X \) is *not* a geometric random variable** because the marginal distribution \( P(X = n) \) does not have the form \( q(1-q)^{n-1} \) for any fixed \( q \).
- The distribution of \( X \) is more complex due to the dependence on the uniformly distributed random variable \( U \), which results in a mixture of geometric distributions with varying success probabilities.

Would you like more details or have any other questions?

Here are 5 related questions:
1. How would you describe the distribution of \( X \) in more detail?
2. What is the effect of the uniform variable \( U \) on the distribution of \( X \)?
3. How does the mixture of geometric distributions affect the properties of \( X \)?
4. What is the expected value \( \mathbb{E}[X] \) given the uniform mixture?
5. How would you compute the variance of \( X \) in this setting?

**Tip:** When dealing with mixtures of distributions, the resulting distribution often has characteristics that differ significantly from any of the individual components.

Explore the analysis of whether X is a geometric random variable with a uniform distribution. Understand the probability theory behind X and its relationship with the uniform variable U. Detailed solution and exploration of the marginal distribution of X provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation