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Question

IMG_20251224_125722_075.jpgIMG_20251224_125722_075.jpg

Реши задачу

Python

python
from PIL import Image, ImageEnhance
img=Image.open('/mnt/data/IMG_20251224_125722_075.jpg')
img.size


(960, 1280)

Python

python
import matplotlib.pyplot as plt, numpy as np
plt.figure(figsize=(7,9))
plt.imshow(img)
plt.axis('off')
plt.show()

Python

python
import numpy as np, matplotlib.pyplot as plt
arr=np.array(img)
# crop top region where text located
crop=Image.fromarray(arr[60:350, 60:900])
plt.figure(figsize=(10,4))
plt.imshow(crop)
plt.axis('off')
plt.show()

Réflexion

Clarifying compound Poisson process and its trajectory

The user is likely asking about the typical trajectory of a compound Poisson process, where S(t)=∑k=1N(t)XkS(t) = \sum_{k=1}^{N(t)} X_k. Here, N(t)N(t) is a non-homogeneous Poisson process with rate function λ(t)\lambda(t), and XkX_k is symmetrically distributed with steps ±2\pm 2. The intensity function λ(t)=3t\lambda(t) = 3t for t≤1t \leq 1 and constant for t>1t > 1.

Clarifying intensity function and trajectory for compound Poisson process

The user is considering a graph with labels that might be confusing. The y-axis could actually represent the intensity function λ(t)\lambda(t) rather than cumulative intensity Λ(t)\Lambda(t), which would explain the linear increase and flat behavior from 0 to 3. The compound Poisson process will have jumps with ±2 equally likely, with arrival times more dense after t≥1t \geq 1.

Building a typical trajectory for compound Poisson process

The user is looking to construct a graph for a typical realization of a compound Poisson process, where jumps are random steps of size ±2. The underlying process has a random intensity function, with the number of jumps governed by a non-homogeneous Poisson process. To solve, I would simulate the number of jumps and sample the jump sizes accordingly. I could also describe the process textually and illustrate it with a step function.

Considering distribution and martingale properties of compound Poisson process

The user is exploring whether the compound Poisson process could be expressed analytically. Since the jumps are symmetric ±2, the process is a martingale with zero drift. They think N(t)N(t) has a cumulative intensity function Λ(t)\Lambda(t), possibly a compensator. The expected arrivals stop after t=1t=1, making cumulative Λ(t)\Lambda(t) piecewise.

Describing typical trajectory of compound Poisson process

The user is confirming that the graph matches the idea of cumulative intensity Λ(t)\Lambda(t), stopping after time 1. This suggests arrivals only occur before 1, with N(t)N(t) being a Poisson process time-changed by Λ(t)\Lambda(t). After time 1, the process stops, and the trajectory becomes a random step function, with jumps of ±2 up to t=1t=1, then remaining constant.

Creating a typical trajectory for compound Poisson process

The task asks to plot the typical trajectory of S(t)S(t), which likely refers to the expected sample path of a compound Poisson process, where arrivals occur based on rate Λ(t)\Lambda(t). We remember that the cumulative intensity, Λ(t)\Lambda(t), follows a linear ramp up to 3 for t≤1t \leq 1 and stays constant afterward. The execution can be clarified by defining N(t)N(t) as Poisson distributed, with specific variance calculations that would help in plotting the path.

Clarifying execution of typical trajectory plot

The task asks to plot a typical trajectory of a compound Poisson process. This typically means showing a stepwise path with jumps influenced by a Poisson count process, where each jump takes values of ±2. I'll aim to demonstrate this with jumps at random times within [0,1], followed by a constant value after the jumps stop at t=1. I'll explain how we arrive at the expected number of jumps in the process and offer a simplified ASCII example.

Thought for 1m 46s

Réponse

По рисунку задан обобщённый пуассоновский (составной) процесс

S(t)=∑k=1N(t)Xk,P{Xk=2}=P{Xk=−2}=12,S(t)=\sum_{k=1}^{N(t)}X_k, \qquad P\{X_k=2\}=P\{X_k=-2\}=\tfrac12,

а функция Λ(t)\Lambda(t) (параметр пуассоновского счётчика) такая:

Λ(t)={3t,0≤t≤1,3,t≥1.\Lambda(t)= \begin{cases} 3t, & 0\le t\le 1,\\[4pt] 3, & t\ge 1. \end{cases}

1) Как выглядит N(t)N(t)

N(t)∼Pois(Λ(t)).N(t)\sim \text{Pois}(\Lambda(t)).

Значит:

  • при 0≤t≤10\le t\le 1: N(t)∼Pois(3t)N(t)\sim \text{Pois}(3t) (в среднем 3t3t скачков),
  • при t≥1t\ge 1: Λ(t)\Lambda(t) больше не растёт ⇒ новых событий нет, поэтому N(t)=N(1),N(1)∼Pois(3).N(t)=N(1),\quad N(1)\sim \text{Pois}(3).

То есть счётчик делает несколько скачков только на отрезке [0,1][0,1], а дальше “замирает”.

2) Как выглядит типичная траектория S(t)S(t)

Процесс S(t)S(t) — ступенчатая функция:

  • старт: S(0)=0S(0)=0,
  • в моменты прихода событий τ1,τ2,…,τN(1)∈(0,1]\tau_1,\tau_2,\dots,\tau_{N(1)}\in(0,1] происходят скачки на ±2\pm 2,
  • после t=1t=1 значение постоянно: S(t)=S(1)S(t)=S(1).

Так как E[N(1)]=3E[N(1)]=3, “типично” увидеть около 3 скачков до t=1t=1. Например, одна возможная траектория (пример!):

  • τ1=0.20,  X1=+2\tau_1=0.20,\; X_1=+2
  • τ2=0.55,  X2=−2\tau_2=0.55,\; X_2=-2
  • τ3=0.87,  X3=+2\tau_3=0.87,\; X_3=+2

Тогда график S(t)S(t) будет таким (ступеньки):

  • S(t)=0S(t)=0 на [0,0.20)[0,0.20)
  • S(t)=2S(t)=2 на [0.20,0.55)[0.20,0.55)
  • S(t)=0S(t)=0 на [0.55,0.87)[0.55,0.87)
  • S(t)=2S(t)=2 на [0.87,1][0.87,1] и дальше всё время S(t)=2S(t)=2.

Схематично:

text
S(t)
 2 |        ┌───────┐     ┌──────────────
 0 |────────┘       └─────┘
-2 |
    0      0.20    0.55  0.87   1        t
                         (после 1 — константа)

Если нужно, могу нарисовать сразу несколько “типичных” траекторий (для разных ω) или посчитать, например, P(N(1)=k)=e−33kk!P(N(1)=k)=e^{-3}\frac{3^k}{k!} и какие уровни S(1)S(1) наиболее вероятны.

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