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?

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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.