You’re absolutely right again — thanks for your patience. Let’s fully align with Obsidian’s markdown LaTeX syntax, using $$ to wrap all math blocks, not \begin{align} directly unless it’s wrapped in $$.
Here’s the fully correct version with proper Obsidian-compatible math blocks:
Memoryless Property – Exponential Distribution
- Definition:
If , then:
P(X>s+t∣X>s)=P(X>t),∀s,t≥0P(X > s + t \mid X > s) = P(X > t), \quad \forall s, t \geq 0
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Explanation:
The probability that the process continues for at least another units of time, given it has already lasted , is the same as the original probability of lasting . -
PDF (Probability Density Function):
fX(x)={λe−λx,x≥00,x<0f_X(x) = \begin{cases} \lambda e^{-\lambda x}, & x \geq 0 \ 0, & x < 0 \end{cases}
- CDF (Cumulative Distribution Function):
- Proof Sketch:
P(X>s+t∣X>s)=P(X>s+t)P(X>s)=e−λ(s+t)e−λs=e−λt=P(X>t)\begin{aligned} P(X > s + t \mid X > s) &= \frac{P(X > s + t)}{P(X > s)} \\ &= \frac{e^{-\lambda(s + t)}}{e^{-\lambda s}} \\ &= e^{-\lambda t} = P(X > t) \end{aligned}
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Key Insight:
The exponential distribution is memoryless — it “forgets” how much time has passed. It’s the only continuous distribution with this property. -
Applications:
Models waiting times between random events in a Poisson process, such as:-
Time between radioactive decays
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Time between customer arrivals
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Time until hardware failure
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Let me know if you’d like a companion card for the geometric distribution, or a visual diagram for intuition!