Approximations for Discrete Distributions

Approximations for discrete distributions are essential tools in the statistics and probability theory that help simplify complex problems involving the discrete random variables. This article aims to provide the detailed overview of these approximations including their relevance, applications and methods.

What are Discrete Distributions?

Discrete distributions describe the probability distribution of discrete random variables, which are variables that take on distinct, countable values.

Unlike continuous distributions, where variables can take any value within a range, discrete distributions only assign probabilities to specific, isolated points (like integers or finite sets of values).

Approximations for Discrete Distributions

Approximations for discrete distributions are used when exact calculations are difficult or impractical. There are several ways to approximate discrete distributions using continuous or other discrete distributions under certain conditions.

The Several methods are used to the approximate discrete distributions. These methods simplify calculations and provide the insights into the behavior of the discrete random variables.

Normal Approximation to the Binomial Distribution

The normal approximation is used when dealing with the binomial distributions especially when the number of trials is large. According to the Central Limit Theorem a binomial distribution B(n,p) can be approximated by the normal distribution N(μ,σ²)

where:

μ = np σ²= np(1−p)

Example: The approximate the binomial probability P(X≤50) for the X∼B(100,0.5).

Solution:

• Mean μ = 100 × 0.5 = 50
• Variance σ² = 100 × 0.5 × 0.5 = 25
• Standard deviation σ = √25 = 5
• Use the normal distribution N(50,25) to the approximate the probability.

Poisson Approximation to the Binomial Distribution

The Poisson approximation is useful when dealing with the binomial distribution where the number of the trials n is large and the probability of success p is small. The Poisson distribution Poisson(λ) can approximate the binomial distribution where λ=np.

Example: The Approximate the probability of the 3 successes in the binomial distribution B(100,0.05) using the Poisson distribution.

Solution:

• λ = 100 × 0.05 = 5
• Use the Poisson distribution Poisson(5) to the approximate the probability of the 3 successes.

Geometric Distribution Approximation

The geometric distribution describes the number of the trials until the first success. When the probability of the success is small the geometric distribution can be approximated using the different distribution based on the problem context.

Example: The Approximate the probability that the first success occurs on the 10th trial for the p=0.1.

Solution:

• The geometric distribution is Geom(p) where p=0.1.
• Use the geometric probability formula to the find the approximation.

Solved Examples on Approximations for Discrete Distributions

Example 1: The Approximate the probability P(X \leq 45) for the X \sim B(100, 0.4) using the normal distribution.

Solution:

Parameters Calculation:

• Mean \mu = n \cdot p = 100 \times 0.4 = 40
• Variance \sigma^2 = n \cdot p \cdot (1 - p) = 100 \times 0.4 \times 0.6 = 24
• Standard deviation \sigma = \sqrt{24} \approx 4.899

Normal Approximation:

• Convert X \leq 45 to the standard normal variable Z: Z = \frac{45 - \mu}{\sigma} = \frac{45 - 40}{4.899} \approx 1.02
• Use the standard normal table to find P(Z \leq 1.02) which is approximately 0.8461.

Example 2: The Approximate the probability of the 4 successes in the binomial distribution B(200, 0.02) using the Poisson distribution.

Solution:

• Poisson Parameter Calculation: \lambda = n \cdot p = 200 \times 0.02 = 4
• Poisson Approximation: Use the Poisson probability formula to find \( P(X = 4) \): P(X = 4) = \frac{e^{-\lambda} \lambda^4}{4!} = \frac{e^{-4} \cdot 4^4}{24} \approx 0.195

Example 3: The Approximate the probability that the first success occurs on the 8th trial for the p = 0.15 using the geometric distribution.

Solution:

The Geometric Probability Calculation:

• The probability \( P(X = 8) \) is given by: P(X = 8) = (1 - p)^{k - 1} \cdot p = (1 - 0.15)^7 \cdot 0.15 \approx 0.15 \times 0.196 = 0.029

Example 4: The Approximate the probability of the getting the 5th success on the 12th trial for the p = 0.25 using the negative binomial distribution.

Solution:

The Negative Binomial Probability Calculation:

• The probability \( P(X = 12) \) is given by: P(X = 12) = \binom{11}{4} \cdot (0.25)^5 \cdot (0.75)^7 \approx 330 \times 0.000976 \times 0.133 = 0.042

Example 5: The Approximate the probability of the drawing 3 white balls in the sample of the 10 from an urn containing 20 white and 30 black balls using the hypergeometric distribution.

Solution:

The Hypergeometric Probability Calculation:

• Use the hypergeometric formula: P(X = 3) = \frac{\binom{20}{3} \cdot \binom{30}{7}}{\binom{50}{10}} \approx \frac{1140 \cdot 2035800}{10272278170} \approx 0.223

Practice Questions on Approximations for Discrete Distributions

Q1. For a binomial distribution B(50,0.6) approximate the probability P(X≥30) using the normal approximation.

Q2. Using the Poisson approximation find the probability of 6 successes in a binomial distribution B(150,0.03).

Q3. Approximate the probability of the first success occurring on the 5th trial for p=0.25 using the geometric distribution.

Q4. For a negative binomial distribution approximate the probability of getting the 7th success on the 15th trial when p=0.2.

Q5. Using the normal approximation, find the probability P(X≤10) for X∼B(25,0.4).

Q6. Approximate the probability of drawing 4 white balls from an urn containing 15 white and 35 black balls in a sample of 12 using the hypergeometric distribution.

Q7. For a binomial distribution B(80,0.5) approximate the probability P(X=40) using the normal approximation.

Q8. Using the Poisson approximation find the probability of 3 successes in a binomial distribution B(80,0.1).

Q9. Approximate the probability of the first success occurring on the 12th trial for p=0.1 using the geometric distribution.

Q10. For a negative binomial distribution approximate the probability of getting the 4th success on the 10th trial with p=0.4.

Conclusion

The Approximations for the discrete distributions are powerful tools that facilitate the handling of the complex statistical problems. By understanding and applying these approximations students and practitioners can streamline their calculations and gain valuable insights into the discrete random variables.