Uniform Distribution — $ \mathrm{Unif}(a,b) $
Definition
Parameters:
- Lower bound $ a \in \mathbb{R} $
- Upper bound $ b > a $
Support:
\(a \le x \le b\)
PDF:
\(f(x \mid a,b)
=
\frac{1}{b-a}\,
\mathbb{1}_{[a,b]}(x)\)
CDF:
\(F(x)
=
\begin{cases}
0, & x < a \\
\frac{x-a}{b-a}, & a \le x \le b \\
1, & x > b
\end{cases}\)
Moment Generating Function (MGF) and Characteristic Function
MGF:
\(M_X(t)
=
\frac{e^{tb}-e^{ta}}{t(b-a)}, \quad t \ne 0\)
Characteristic Function:
\(\varphi_X(t)
=
\frac{e^{itb}-e^{ita}}{it(b-a)}\)
Moments
Mean and Variance
\(\mathbb{E}[X] = \frac{a+b}{2}\)
\[\operatorname{Var}(X) = \frac{(b-a)^2}{12}\]Raw Moments
\(\mathbb{E}[X^k] = \frac{b^{k+1}-a^{k+1}}{(k+1)(b-a)}\)
Location–Scale Family
If $ X \sim \mathrm{Unif}(0,1) $, then \(a+(b-a)X \sim \mathrm{Unif}(a,b)\)
Thus Uniform is a location–scale family.
Order Statistics
Let $ X_1,\dots,X_n \stackrel{iid}{\sim} \mathrm{Unif}(a,b) $.
Minimum
\(X_{(1)} \sim a + (b-a)\,\mathrm{Beta}(1,n)\)
Maximum
\(X_{(n)} \sim a + (b-a)\,\mathrm{Beta}(n,1)\)
General Order Statistic
\(\frac{X_{(k)}-a}{b-a} \sim \mathrm{Beta}(k,n-k+1)\)
Sufficient Statistics
For $ \mathrm{Unif}(a,b) $ with both parameters unknown:
-
Minimal sufficient statistic: \((X_{(1)}, X_{(n)})\)
-
Not complete
This is a classic nonregular family.
Likelihood and Estimation
Likelihood
\(L(a,b) = (b-a)^{-n} \mathbb{1}\{a \le X_{(1)},\, X_{(n)} \le b\}\)
MLEs
\(\hat{a} = X_{(1)}, \quad \hat{b} = X_{(n)}\)
Bias:
\(\mathbb{E}[X_{(1)}] = a + \frac{b-a}{n+1}\)
\(\mathbb{E}[X_{(n)}] = b - \frac{b-a}{n+1}\)
UMVU Estimators
Using order statistics:
\[\tilde{a} = X_{(1)} - \frac{X_{(n)}-X_{(1)}}{n-1}\] \[\tilde{b} = X_{(n)} + \frac{X_{(n)}-X_{(1)}}{n-1}\]Exist despite lack of completeness.
Nonregularity
- Support depends on parameters
- Fisher information not well-defined
- CRLB does not apply
Exam favorite: counterexample to standard theory assumptions.
Transformations
Probability Integral Transform
If $ X \sim \mathrm{Unif}(a,b) $, then \(\frac{X-a}{b-a} \sim \mathrm{Unif}(0,1)\)
Conversely, if $ U \sim \mathrm{Unif}(0,1) $, then \(F^{-1}(U) \sim F\)
Bayesian Notes
- No natural conjugate prior for $ (a,b) $
- Often handled via hierarchical or improper priors
Key Theorems and Facts (Prelim-Relevant)
- Order statistic distributions
- Minimal but not complete sufficiency
- Nonregular family
- Failure of CRLB
- Probability integral transform
Exam Takeaways
- Always reduce to $ \mathrm{Unif}(0,1) $
- Order statistics carry all information
- Expect boundary-based estimators
- Uniform breaks many “nice” theorems
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