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Topics Covered

PDF CDF Moments MGF / CF / CGF Skewness Kurtosis Mean Deviation
On this page
  1. 1. Definition & PDF
  2. 2. Distribution Function (CDF)
  3. 3. Moments via MGF
  4. 4. Other Generating Functions
  5. 5. Skewness & Kurtosis
  6. 6. Mean Deviation about Mean
  7. 7. Worked Examples
  8. Key Take-aways

1. Definition & PDF

DEFINITION

A continuous random variable \(X\) is said to follow a Uniform distribution on the interval \((a, b)\), \(a < b\), if its PDF is constant on that interval and zero outside:

\[ f(x) \;=\; \begin{cases} \dfrac{1}{b - a}, & a \le x \le b\\[4pt] 0, & \text{otherwise}. \end{cases} \]

Notation: \(X \sim U(a, b)\). Every value in \((a,b)\) is equally likely.

Validity: \(\int_a^b \dfrac{1}{b-a}\,dx = \dfrac{b-a}{b-a} = 1\) ✓.

a b 1/(b−a) PDF f(x)
Constant on [a, b]
a b 1 CDF F(x)
Linear ramp from 0 to 1

2. Distribution Function (CDF)

\[ F(x) \;=\; \begin{cases} 0, & x < a\\ \dfrac{x - a}{b - a}, & a \le x \le b\\ 1, & x > b. \end{cases} \]

3. Moments via MGF

MGF \[ M_X(t) \;=\; \int_a^b \dfrac{e^{tx}}{b - a}\,dx \;=\; \dfrac{e^{tb} - e^{ta}}{t(b - a)}, \quad t \ne 0. \]

Mean & Variance

\[ E(X) \;=\; \dfrac{a + b}{2}, \qquad \text{Var}(X) \;=\; \dfrac{(b - a)^2}{12}. \]

Derivation of mean: \(E(X) = \int_a^b \dfrac{x}{b-a}\,dx = \dfrac{1}{b-a}\cdot \dfrac{b^2 - a^2}{2} = \dfrac{a+b}{2}\).

Variance: \(E(X^2) = \dfrac{a^2 + ab + b^2}{3}\). Then Var \(= E(X^2) - [E(X)]^2 = \dfrac{(b-a)^2}{12}\).

4. Other Generating Functions

\[ \phi_X(t) \;=\; \dfrac{e^{itb} - e^{ita}}{it(b - a)}, \qquad K_X(t) \;=\; \ln M_X(t). \]

5. Skewness & Kurtosis

\[ \mu_3 = 0, \qquad \mu_4 = \dfrac{(b-a)^4}{80}. \] \[ \beta_1 = 0, \qquad \beta_2 = \dfrac{\mu_4}{\mu_2^2} = \dfrac{(b-a)^4/80}{(b-a)^4/144} = \dfrac{144}{80} = 1.8. \]

Symmetric (\(\beta_1 = 0\)); strongly platykurtic since \(\beta_2 = 1.8 < 3\). \(\gamma_2 = -1.2\).

6. Mean Deviation about Mean

For \(X \sim U(a,b)\), the mean is \(\mu = (a+b)/2\). Then

\[ \text{MD}_\mu \;=\; E|X - \mu| \;=\; \dfrac{b - a}{4}. \]

Derivation: By symmetry, with \(b - \mu = (b-a)/2\), \(E|X - \mu| = 2\displaystyle\int_\mu^b (x - \mu) \dfrac{1}{b-a}\,dx = \dfrac{(b-\mu)^2}{b-a} = \dfrac{\big[(b-a)/2\big]^2}{b-a} = \dfrac{b-a}{4}\).

7. Worked Examples

EXAMPLE 1

Bus arrives every 30 minutes; arrival time after a person reaches the stop is \(U(0, 30)\). Find the probability of waiting (i) at most 5 minutes, (ii) more than 20 minutes.

(i) \(P(X \le 5) = 5/30 = 1/6\).
(ii) \(P(X > 20) = (30-20)/30 = 1/3\).
Mean wait = 15 min; SD = \(30/\sqrt{12} \approx 8.66\) min.

EXAMPLE 2

\(X \sim U(0, 10)\). Find the MD about mean and variance.

Mean = 5, Variance = 100/12 ≈ 8.33; SD ≈ 2.886.

MD = (10 − 0)/4 = 2.5.

Key Take-aways

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