Estimation of Lower Bound in Poisson Distribution

Normal Distribution Explorer

Visualize the Bell Curve and calculate probabilities.

Mean (μ)

Standard Deviation (σ)

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Find Probability: P(X < x) P(X > x) P(a < X < b)

Value (x)

Lower (a)

Upper (b)

Probability: 0.0000

Theory: The Normal Distribution

Topic: Continuous Distributions | Difficulty: Intermediate

The Normal Distribution (or Gaussian Distribution) is the most important continuous probability distribution in statistics. It is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

Probability Density Function (PDF)

`f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}`

Where $-\infty < x < \infty$, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Moments

• Mean ($E[X]$): $\mu$
• Variance ($Var(X)$): $\sigma^2$
• Standard Deviation: $\sigma$

Shape Parameters

• Skewness ($\beta_1$): 0 (Symmetric)
• Kurtosis ($\beta_2$): 3
• Excess Kurtosis: 0 (Mesokurtic)

Quartiles & Percentiles

Due to symmetry, the quartiles are equidistant from the mean:

`Q_1 \approx \mu - 0.6745\sigma \quad \text{(25th Percentile)}` `Q_2 = \mu \quad \text{(Median / 50th Percentile)}`

`Q_3 \approx \mu + 0.6745\sigma \quad \text{(75th Percentile)}`

Python Implementation

Python (Scipy)

import scipy.stats as stats

mu = 0
sigma = 1
x = 1.96

# 1. Calculate Probability P(X < x) (CDF)
prob = stats.norm.cdf(x, mu, sigma)
print(f"P(X < {x}) = {prob:.4f}")

# 2. Find Percentile (Inverse CDF / PPF)
# Find x such that P(X < x) = 0.95
crit_val = stats.norm.ppf(0.95, mu, sigma)
print(f"95th Percentile = {crit_val:.4f}")