Probability and Statistics Notes

Continuous Random Variables

Waiting Time between Successive Speeders (Exercise 3.14)

• Definition: Waiting time in hours is modeled by a continuous random variable with cumulative distribution function.
• Cumulative Distribution Function (CDF):
  F(x) = \begin{cases} 0, & \text{if } x < 0 \
  1 - e^{-4x}, & \text{if } x \geq 0 \end{cases}
• Goals:
  • (a) Calculate the probability of waiting less than 12 minutes (0.2 hours) using the CDF:
  • $P(X < 0.2) = F(0.2)$
  • (b) Calculate using the probability density function (PDF):
  • $f(x) = 4e^{-4x}$
  • $P(X < 0.2) = \int_0^{0.2} 4e^{-4x} dx$

Uniform Density between 1 and 3 (Exercise 3.17)

• Properties: Continuous random variable, density function defined between x = 1 and x = 3,
• PDF:
  f(x) = \begin{cases} \frac{1}{2}, & 1 \leq x \leq 3 \
  0, & \text{elsewhere} \end{cases}
• (a) Area under the curve must equal 1:
  • $\int_1^{3} \frac{1}{2} dx = 1$
• (b) Probability calculation:
  • $P(2 < x < 2.5) = \int_2^{2.5} \frac{1}{2} dx$
• (c) Cumulative probability:
  • $P(X \leq 1.6) = \int_{1}^{1.6} \frac{1}{2} dx$

Particle Size Distribution (Exercise 3.29)

• Density Function: Defined for particle sizes in micrometers,
• PDF:
  f(x) = \begin{cases} 3x - 4 & x > 1 \
  0 & \text{elsewhere} \end{cases}
• Verify Valid Density:
  • Check conditions for a probability density function.
• CDF Calculation:
  • $F(x) = \int_1^x (3t-4) dt$
• Probability Calculation:
  • $P(X > 4) = 1 - F(4)$

Measurement Error Distribution (Exercise 3.30)

• PDF: For measurement error with bounds,
• Function:
  f(x) = \begin{cases} k(3 - x^2), & -1 \leq x \leq 1 \
  0, & \text{elsewhere} \end{cases}
• (a) Find normalization constant k:
  • Set the integral across the domain equal to 1.
• (b) Probability of error < 1/2, using the PDF:
  • 
    $P(X < 1/2) = \int_{-1}^{1/2} k(3 - x^2) dx$
• (c) Magnitude probability scaling:
  • Examine the region $|x| > 0.8$ to compute.

Joint Probability Distribution of X and Y (Exercise 3.38)

• Joint PDF:
  $f(x, y) = \frac{x + y}{30}, \quad \text{for } x = 0,1,2,3; y = 0,1,2$
• Tasks:
  • (a) Compute various probabilities using the joint distribution to answer:
  • $P(X \leq 1, Y = 1)$
  • $P(X > 1, Y < 1)$
  • $P(X \leq Y)$
  • $P(X + Y = 2)$

Measurement Density Function under Specific Conditions (3.36)

• Given Function:
  f(x) = \begin{cases} 2(1-x), & 0 < x < 1 \
  0, & \text{elsewhere} \end{cases}
• Probabilities to compute:
  • (a) $P(X \leq 0.5)$
  • (b) $P(X > 0.4)$
  • (c) Conditional probability when $X \geq 0.5$

Pressure Variability in Airplane Tires (Exercise 3.44)

• Pressure Random Variables:
  • Let X and Y be independent tire pressures.
  • Joint PDF:
    $f(x, y) = k(x^2 + y^2)$
• Tasks:
  • Determine k such that the function integrates to 1.
  • Compute probabilities for different ranges of tire pressures.

Miscellaneous Probability & CDF Problems

• Shelf Life of Bakery Items (3.71): Use density function to evaluate probabilities of sellability.
• Passenger Congestion on Airport Trains (3.72): Analyze a uniform distribution on the travel time.