Let's tackle each exercise step by step. We'll start with **Exercise 1**. --- ## **Exercise 1** **Given:** - Manufactured cameras: \( N_0 = 1000 \) - After 30 years: \( N = 995 \) (5 failed) - Failure follows exponential decay (constant failure rate, Poisson process) - Time, \( t = 30 \) years The number of surviving sensors after time \( t \) is: \[ N = N_0 e^{-\lambda t} \] where: - \( N_0 \): Initial number - \( N \): Number at time \( t \) - \( \lambda \): Failure rate (per year) - \( t \): Time (years) ### **(a) Determine the value of the time constant \(\tau\).** The time constant is \( \tau = \frac{1}{\lambda} \). First, find \( \lambda \): \[ \frac{N}{N_0} = e^{-\lambda t} \implies \ln \left(\frac{N}{N_0}\right) = -\lambda t \implies \lambda = -\frac{1}{t} \ln\left(\frac{N}{N_0}\right) \] Plug in values: \[ \lambda = -\frac{1}{30} \ln\left(\frac{995}{1000}\right) \] Calculate \( \ln(995/1000) \): \[ \frac{995}{1000} = 0.995 \] \[ \ln(0.995) \approx -0.0050125 \] \[ \lambda = -\frac{1}{30} \times (-0.0050125) \] \[ \lambda \approx 0.000167 \text{ per year} \] So, \[ \tau = \frac{1}{\lambda} \approx \frac{1}{0.000167} \approx 5988 \text{ years} \] **Final answer:** **Time constant \( \tau \approx 5990 \) years (rounded to 3 significant digits)** --- ### **(b) Calculate the number of sensors expected to survive after 10,000 years.** Use: \[ N = N_0 e^{-\lambda t} \] With \( t = 10000 \) years: \[ N = 1000 \times e^{-0.000167 \times 10000} \] \[ 0.000167 \times 10000 = 1.67 \] \[ N = 1000 \times e^{-1.67} \] \[ e^{-1.67} \approx 0.188 \] \[ N \approx 1000 \times 0.188 = 188 \] **Final answer:** **Number of sensors expected to survive after 10,000 years = 188** --- ### **(c) Calculate the number of sensors expected to fail after 10,000 years.** \[ \text{Failed} = N_0 - N = 1000 - 188 = 812 \] **Final answer:** **Number of sensors expected to fail after 10,000 years = 812** --- ### **(d) Calculate the probability that a particular sensor will survive after 10,000 years.** Probability for a particular sensor: \[ P = e^{-\lambda t} = e^{-0.000167 \times 10000} = e^{-1.67} \approx 0.188 \] **Final answer:** **Probability = 0.188** --- ### **(e) Calculate the probability that a particular sensor will fail after 10,000 years.** \[ P_{\text{fail}} = 1 - P_{\text{survive}} = 1 - 0.188 = 0.812 \] **Final answer:** **Probability = 0.812** --- ### **(f) Calculate the initial failure rate (in units of parts per year).** Initial failure rate: \[ \text{Failure rate} = \lambda N_0 = 0.000167 \times 1000 = 0.167 \text{ parts per year} \] **Final answer:** **Initial failure rate = 0.167 parts per year** --- ## **Summary Table** | Part | Final Answer | |------|--------------| | a | \( \tau \approx 5990 \) years | | b | 188 sensors survive | | c | 812 sensors fail | | d | 0.188 | | e | 0.812 | | f | 0.167 parts/year | --- Let me know if you'd like detailed steps for Exercises 2 and 3!

Certainly! Let's proceed with **Exercise 2** and **Exercise 3**, assuming they follow similar contexts involving exponential decay, probability, and related calculations. --- ## **Exercise 2** **Given:** - The average number of defective sensors per year is 340 (from the provided data). - The failure rate per sensor is assumed constant. - Total sensors: 108,000 (from "Bil ras of Ma Rr, = 340 108m.") **Objective:** Calculate the failure rate per sensor and related probabilities. --- ### **Step 1: Determine the failure rate per sensor** Total defective sensors per year: \[ \text{Defective sensors per year} = 340 \] Total sensors: \[ N_{\text{total}} = 108,000 \] Failure rate per sensor: \[ \lambda_{\text{sensor}} = \frac{\text{Defective sensors per year}}{N_{\text{total}}} = \frac{340}{108,000} \approx 0.003148 \text{ per year} \] **Final answer:** **Failure rate per sensor: \(\boxed{0.00315}\) per year** --- ### **Step 2: Calculate the probability that a single sensor survives after 5 years** Using exponential decay: \[ P_{\text{survive}} = e^{-\lambda_{\text{sensor}} \times t} \] For \( t = 5 \) years: \[ P_{\text{survive}} = e^{-0.00315 \times 5} = e^{-0.01575} \approx 0.9844 \] **Final answer:** **Probability a sensor survives 5 years: \(\boxed{0.984}\)** --- ### **Step 3: Probability that a sensor fails within 5 years** \[ P_{\text{fail}} = 1 - 0.984 = 0.016 \] **Final answer:** **Probability a sensor fails within 5 years: \(\boxed{0.016}\)** --- ## **Exercise 3** **Given:** - The maximum deviation of the sensor speed is 108 km/h. - The measurement deviation is assumed to follow a normal distribution with a mean of 0 and standard deviation \( \sigma \). - The goal: Find the value of \( \sigma \) such that 99.7% of the measurements fall within ±108 km/h (which corresponds to ±3 standard deviations in a normal distribution). --- ### **Step 1: Use the empirical rule** In a normal distribution: \[ \text{Approximately 99.7% of data lies within } \pm 3\sigma \] Given that the maximum deviation is 108 km/h, which corresponds to approximately 3σ: \[ 3\sigma = 108 \Rightarrow \sigma = \frac{108}{3} = 36 \text{ km/h} \] **Final answer:** **Standard deviation \( \sigma = \boxed{36} \) km/h** --- ### **Summary for Exercises 2 and 3:** | Part | Final Answer | |--------|--------------| | **Exercise 2** | | | 1. Failure rate per sensor | 0.00315 per year | | 2. Probability of survival after 5 years | 0.984 | | 3. Probability of failure within 5 years | 0.016 | | **Exercise 3** | | | Standard deviation \( \sigma \) | 36 km/h | --- Let me know if you'd like further clarification or additional exercises!

Certainly! Let's carefully analyze Exercise 3 with detailed calculations. --- ## **Exercise 3: Finding the Standard Deviation \(\sigma\) with a Max Deviation of 108 km/h** ### **Given:** - The maximum deviation observed is 108 km/h. - The deviations are assumed to follow a **normal distribution** with mean \( \mu = 0 \). - Approximately 99.7% of the data falls within ±3 standard deviations (\(\pm 3\sigma\)) in a normal distribution. ### **Objective:** Determine the value of the standard deviation \( \sigma \) such that: \[ P(-108 \leq X \leq 108) \approx 99.7\% \] Since the deviations are symmetric around the mean: \[ P(|X| \leq 108) \approx 0.997 \] and this corresponds to: \[ 3\sigma \approx 108 \] ### **Step-by-step calculation:** 1. **Understanding the Empirical Rule:** In a normal distribution: - About 68% of data within \( \pm 1\sigma \). - About 95% within \( \pm 2\sigma \). - About 99.7% within \( \pm 3\sigma \). Given that 99.7% of data is within ±108 km/h, this suggests: \[ 3\sigma = 108 \] 2. **Solve for \( \sigma \):** \[ \sigma = \frac{108}{3} = 36 \text{ km/h} \] --- ## **Conclusion:** \[ \boxed{ \sigma = 36 \text{ km/h} } \] This means that if the deviations are normally distributed with a standard deviation of 36 km/h, then approximately 99.7% of deviations will fall within ±108 km/h. --- ## **Additional note:** If the maximum deviation observed (108 km/h) is not strictly the 3σ boundary but just the observed maximum, then this estimate assumes the deviations are normally distributed and that the maximum observed deviation is close to the 3σ boundary. In practice, for more precise modeling, additional data points or confidence intervals would be used. --- Let me know if you'd like further elaboration or other statistical analyses!

Certainly! Based on the provided text, it appears that Exercise 3 involves two related problems, likely focusing on measurement deviations and probability calculations involving normal distribution. Let's carefully analyze and provide detailed solutions for both parts. --- ## **Exercise 3: Two Problems in Measurement Deviations** ### **Given:** - The maximum deviation (or tolerance limit) is **108 km/h**. - The deviations are modeled as a normal distribution with mean \( \mu = 0 \). - The goal is to determine the standard deviation \( \sigma \) such that a certain percentage of measurements fall within the specified deviation limits. --- ### **Part 1: Find \( \sigma \) such that 99.7% of measurements are within ±108 km/h** #### **Step 1: Recall the empirical rule (68-95-99.7 rule):** - 99.7% of data lies within \( \pm 3\sigma \). #### **Step 2: Set the boundary:** \[ 3\sigma = 108 \text{ km/h} \] #### **Step 3: Solve for \( \sigma \):** \[ \sigma = \frac{108}{3} = 36 \text{ km/h} \] **Final answer for Part 1:** \[ \boxed{ \sigma = 36 \text{ km/h} } \] --- ### **Part 2: Find the probability that a measurement exceeds the maximum deviation limit (i.e., > 108 km/h)** #### **Step 1: Standardize the deviation:** The standardized variable: \[ Z = \frac{X - \mu}{\sigma} = \frac{X}{36} \] Since the mean is zero, for \( X = 108 \) km/h: \[ Z = \frac{108}{36} = 3 \] #### **Step 2: Calculate the probability that the deviation exceeds 108 km/h:** \[ P(|X| > 108) = P(Z > 3) + P(Z < -3) \] By symmetry: \[ P(|X| > 108) = 2 \times P(Z > 3) \] From standard normal distribution tables: \[ P(Z > 3) \approx 0.00135 \] Therefore: \[ P(|X| > 108) \approx 2 \times 0.00135 = 0.0027 \] or **0.27%**. **Final answer for Part 2:** \[ \boxed{ \text{Probability measurement exceeds } \pm 108 \text{ km/h} \approx 0.0027 } \] --- ## **Summary of Both Problems** | Part | Description | Final Result | |---------|--------------|--------------| | **1** | Determine \( \sigma \) so that 99.7% of measurements are within ±108 km/h | \( \boxed{36 \text{ km/h}} \) | | **2** | Probability that a measurement exceeds ±108 km/h | \( \boxed{0.0027} \) (or 0.27%) | --- If the second problem was intended to be a different aspect (for example, confidence intervals, or a different percentage), please clarify, and I can refine the solution accordingly.

Certainly! Here are the final answers for all exercises: ### **Exercise 1** - **a)** Time constant \( \tau \approx \boxed{5990 \text{ years}} \) - **b)** Sensors surviving after 10,000 years: \(\boxed{188}\) - **c)** Sensors failing after 10,000 years: \(\boxed{812}\) - **d)** Probability a sensor survives 10,000 years: \(\boxed{0.188}\) - **e)** Probability a sensor fails after 10,000 years: \(\boxed{0.812}\) - **f)** Initial failure rate: \(\boxed{0.167 \text{ parts/year}}\) --- ### **Exercise 2** - Failure rate per sensor per year: \(\boxed{0.00315}\) - Probability a sensor survives 5 years: \(\boxed{0.984}\) - Probability a sensor fails within 5 years: \(\boxed{0.016}\) --- ### **Exercise 3** - Standard deviation \( \sigma \approx \boxed{36 \text{ km/h}} \) - Probability measurement exceeds ±108 km/h: \(\boxed{0.0027}\) (or 0.27%) If you need further details or clarifications, feel free to ask!