Binomial Approximation Using Poisson Distribution
The Binomial distribution, denoted as $B(n, p)$, describes the number of successes in a fixed number of independent trials ($n$), each with the same probability of success ($p$).
Approximation Conditions
- When the number of trials, $n$, is large.
- And the probability of success, $p$, is small (close to 0).
Approximation Method
Under these specific conditions (large $n$, small $p$), the Binomial distribution can be accurately approximated by the Poisson distribution. The parameter $\lambda$ (lambda) for the Poisson distribution is calculated as the product of $n$ and $p$.
Formula: $\lambda = n \times p$
So, $B(n, p) \approx Poisson(\lambda = np)$
Why Poisson Distribution?
The Poisson distribution is suitable for modeling the number of events occurring in a fixed interval of time or space, given a constant average rate. When $n$ is large and $p$ is small, the Binomial distribution's behavior closely resembles this scenario, with $\lambda$ representing the average number of successes.
Other Distributions
- Normal Distribution: Approximates Binomial when $n$ is large and $p$ is not too close to 0 or 1 ($np > 5$ and $n(1-p) > 5$ are common rules of thumb).
- Hypergeometric Distribution: Used for sampling without replacement from a finite population.
- Bernoulli Distribution: Applies only to a single trial ($n=1$).
Therefore, for a large number of trials ($n$) and a small probability of success ($p$), the Poisson Distribution is the appropriate approximation for the Binomial distribution.
Important Questions from Probability Distribution
If the mean and variance of a binomial distribution are 5 and 4, respectively, then the value of n is:
For the distribution with unknown θ
\(f(x,\theta ) = \left\{ {\begin{array}{*{20}{c}} {\frac{1}{\theta };0 \le x \le \theta }\\ {0;elsewhere} \end{array}} \right.\)
We set the testing of hypothesis H 0 ∶ θ = 1 vs H 1 ∶ θ = 2. When the critical region X ≥ 0.4, the value of probability of type-II error is:
For the cumulative distribution function \(F(x) = \left\{ {\begin{array}{*{20}{c}} {0;x < - 1}\\ {\frac{1}{2}{{(x + 1)}^2}; - 1 \le x < 0}\\ {1 - \frac{{{{(1 - x)}^2}}}{2};0 \le x < 1}\\ {1.1 < x < \infty } \end{array}} \right.\)
the upper quartile point is
Let the joint probability density function of \( (X, Y) \) be
\[f(x, y) = \begin{cases} 6xy^2 & \text{if } 0 < x < 1, 0 < y < 1 \\ 0, & \text{otherwise} \end{cases}\]
Then \( P\left(\frac{1}{2} < X < \frac{3}{4}\right) \) is:
Let X and Y have the joint p.m.f. f(x, y) = x + y / 21, where x = 1, 2, 3 and y = 1, 2. The marginal p.m.f. of X is:
