Math Problem Statement

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

Solution

Part 1: PMF of XX

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

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

Where:

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

Part 2: Truncated Distribution

When the distribution is truncated to the right at x=6x = 6, it means we consider only the values X=0,1,2,,6X = 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=kX6)=P(X=k)P(X6)=(6k)(0.35)k(0.65)6kk=06(6k)(0.35)k(0.65)6kP(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(X6)=1P(X \leq 6) = 1, the PMF remains the same as the original binomial distribution:

P(X=kX6)=(6k)(0.35)k(0.65)6kP(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=6x = 6 doesn't remove any probability mass (as P(X6)=1P(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):

E[X]=np=60.35=2.1\mathbb{E}[X] = n \cdot p = 6 \cdot 0.35 = 2.1

  • Variance:

Var(X)=np(1p)=60.350.65=1.365\text{Var}(X) = n \cdot p \cdot (1 - p) = 6 \cdot 0.35 \cdot 0.65 = 1.365

Summary:

  • The PMF of XX remains (6k)(0.35)k(0.65)6k\binom{6}{k} (0.35)^k (0.65)^{6-k}.
  • The mean of the distribution is 2.12.1.
  • The variance of the distribution is 1.3651.365.

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=2x = 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 pp 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.

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Math Problem Analysis

Mathematical Concepts

Binomial Distribution
Probability Mass Function (PMF)
Truncated Distributions
Mean and Variance

Formulas

Binomial coefficient formula
Mean formula for binomial distribution
Variance formula for binomial distribution

Theorems

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Suitable Grade Level

Advanced High School