x follows bin (6, 0.35). State its pmf. If distribution is truncated to the right at x=6, obtain its pdf and obtain its mean and variance

### Part 1: PMF of \( X \)

Given that \( X \) follows a binomial distribution with parameters \( n = 6 \) and \( p = 0.35 \), the probability mass function (PMF) of \( X \) is given by:

\[
P(X = k) = \binom{6}{k} (0.35)^k (0.65)^{6-k}, \quad k = 0, 1, 2, \ldots, 6
\]

Where:
- \( \binom{6}{k} \) is the binomial coefficient, representing the number of ways to choose \( k \) successes out of 6 trials.

### Part 2: Truncated Distribution

When the distribution is truncated to the right at \( x = 6 \), it means we consider only the values \( X = 0, 1, 2, \ldots, 6 \), but the total probability is rescaled so that it sums to 1 over this range.

The probability mass function (PMF) of the truncated distribution becomes:

\[
P(X = k \mid X \leq 6) = \frac{P(X = k)}{P(X \leq 6)} = \frac{\binom{6}{k} (0.35)^k (0.65)^{6-k}}{\sum_{k=0}^{6} \binom{6}{k} (0.35)^k (0.65)^{6-k}}
\]

Given that \( P(X \leq 6) = 1 \), the PMF remains the same as the original binomial distribution:

\[
P(X = k \mid X \leq 6) = \binom{6}{k} (0.35)^k (0.65)^{6-k}
\]

### Part 3: Mean and Variance

Since truncating at \( x = 6 \) doesn't remove any probability mass (as \( P(X \leq 6) = 1 \)), the mean and variance of the truncated distribution are the same as those of the original binomial distribution:

- **Mean (Expected Value):**

\[
\mathbb{E}[X] = n \cdot p = 6 \cdot 0.35 = 2.1
\]

- **Variance:**

\[
\text{Var}(X) = n \cdot p \cdot (1 - p) = 6 \cdot 0.35 \cdot 0.65 = 1.365
\]

### Summary:

- The PMF of \( X \) remains \( \binom{6}{k} (0.35)^k (0.65)^{6-k} \).
- The mean of the distribution is \( 2.1 \).
- The variance of the distribution is \( 1.365 \).

Do you want further details or have any questions?

---

**Relative Questions:**
1. What is the cumulative distribution function (CDF) of a binomial distribution?
2. How would the PMF change if the distribution was truncated to the left at \( x = 2 \)?
3. How do you calculate the mode of a binomial distribution?
4. What is the significance of truncation in probability distributions?
5. How do you derive the moment generating function of a binomial distribution?
6. How does the variance of a binomial distribution change as \( p \) increases?
7. How does the binomial distribution compare to the Poisson distribution?
8. What are some real-world applications of truncated binomial distributions?

**Tip:** Truncated distributions can be useful in scenarios where certain outcomes are impossible or excluded by design.

Explore the binomial distribution with a focus on its PMF, truncation at x=6, and calculations of mean and variance. Get insights into how truncation affects probability distributions. Suitable for advanced high school students and anyone interested in probability theory.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation