Estimation of Lower Bound in Poisson Distribution

Student's T-Distribution Explorer

Analyze heavy tails and small sample behavior.

Degrees of Freedom (ν) Increase ν to see it become Normal.

Compare with Normal (Z)

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Find Probability: Two-Tailed (P > |t|) Left Tail (P < t) Right Tail (P > t)

t-Score (t)

Probability (Area): 0.0000

Theory: Student's t-Distribution

Topic: Inference | Difficulty: Advanced

The Student's t-distribution plays a crucial role in statistical inference when the sample size is small ($n < 30$) and the population standard deviation ($\sigma$) is unknown. It is symmetric and bell-shaped like the Normal distribution but has heavier tails, meaning it is more prone to producing values that fall far from the mean.

Probability Density Function (PDF)

`f(t) = \frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\nu\pi}\Gamma(\frac{\nu}{2})} \left(1 + \frac{t^2}{\nu} \right)^{-\frac{\nu+1}{2}}`

Where $\nu$ is degrees of freedom and $\Gamma(\cdot)$ is the Gamma function.

Moments

• Mean: 0 (Defined only for $\nu > 1$)
• Variance: $\frac{\nu}{\nu-2}$ (Defined for $\nu > 2$)
• Skewness: 0 (Symmetric, for $\nu > 3$)
• Kurtosis (Excess): $\frac{6}{\nu-4}$ (for $\nu > 4$)

Generating Functions

• MGF: Does Not Exist. Because of the heavy tails, the integral diverges.
• Characteristic Function: Exists.
  $\phi(t) = \frac{K_{\nu/2}(\sqrt{\nu}|t|)\sqrt{\nu}|t|^{\nu/2}}{\Gamma(\nu/2) 2^{\nu/2-1}}$
  (Where $K$ is the Bessel function of 2nd kind)

Relationships & Asymptotics

1. Convergence to Normal: As $\nu \to \infty$, the t-distribution converges to the Standard Normal Distribution $N(0,1)$.
2. Relation to Normal & Chi-Square: If $Z \sim N(0,1)$ and $V \sim \chi^2_\nu$, then: $$ T = \frac{Z}{\sqrt{V/\nu}} \sim t_\nu $$ 3. Relation to F-Distribution: $T^2 \sim F(1, \nu)$. The square of a t-variable follows an F-distribution.

Real-World Applications

1. One-Sample t-Test: Used to test if a population mean is equal to a specified value when $\sigma$ is unknown.
2. Confidence Intervals: Constructing intervals for the mean with small samples ($n < 30$).
3. Regression Analysis: Testing the significance of regression coefficients ($\beta$) in linear models.
4. Finance: Modeling asset returns that exhibit "fat tails" (extreme risks) which the Normal distribution underestimates.

Python Implementation

Python (Scipy)

import scipy.stats as stats

df = 10     # Degrees of Freedom
t_score = 2.228

# 1. Two-Tailed P-Value
# P(|T| > t) = 2 * (1 - CDF(|t|))
p_val = 2 * (1 - stats.t.cdf(abs(t_score), df))
print(f"Two-Tailed P-Value = {p_val:.4f}")

# 2. Critical Value (Inverse CDF)
# Find t for 95% Confidence (alpha = 0.05, two-tailed)
alpha = 0.05
crit_val = stats.t.ppf(1 - alpha/2, df)
print(f"Critical t-value = {crit_val:.4f}")