Name Hw Unit 3 Du

Name Hwunit 3 Du

Please read each question carefully and answer it completely. “Extra work†not explicitly needed can be copied into the end of document in the Appendix. If an answer is incorrect, including work in the Appendix might allow for minimal “partial credit.â€

Consider the following four continuous probability distributions: 1. Uniform (Continuous) on the interval [0,2]; 2. Standard Logistic; 3. Standard Normal; 4. Exponential with lambda = 2. Complete the table with the probability of each event:

  • {0
  • {X
  • {X=0}
  • {0.25

Report any answers to three decimal places. For 1. and 2., use techniques other than distribution commands in EXCEL. For 1., consider geometrical area or the general form of the cdf; for 2., use the cdf function. For 3. and 4., use distribution commands in EXCEL.

Suppose the probability density function for a random variable X equals the following: f(x) = 3x2 for {0

  1. Verify the two conditions for a valid pdf: positivity and total area under the curve equal to 1.
  2. Using integration, find E(X) and the cdf, F(X).
  3. Use F(X) to find P(0.25

In a study on traffic fatalities, the number of fatal crashes is modeled with the normal distribution, with mean 1550 and standard deviation 300. Fill in the table using cdf notation and EXCEL commands for the following questions:

  1. Probability that crashes fewer than 1000: cdf notation: F(1000) or Φ(-1.833) EXCEL: =NORM.DIST(1000,1550,300,TRUE) or =NORM.S.DIST(-1.833,TRUE)
  2. Probability between 1000 and 2000: cdf: F(2000) - F(1000) EXCEL: =NORM.DIST(2000,1550,300,TRUE) - NORM.DIST(1000,1550,300,TRUE)
  3. Number of crashes to be in top 5%: answer: approximately at least 2043 crashes, using inverse cdf: =NORM.INV(0.95,1550,300)

Using the Empirical Rule: fill in approximate probabilities for X between given bounds:

  • 1252 and 1848 for about 68%
  • 962 and 2138 for about 95%
  • 777 and 2323 for about 99%

In analyzing the service efficiency of bank tellers, where the number of customers served per hour follows a Poisson distribution with mean 12, answer the following:

  1. Probability more than 10 customers in an hour: cdf notation: P(X > 10) = 1 - P(X ≤ 10) = 1 - F(10) EXCEL: =1 - POISSON.DIST(10,12,TRUE)
  2. Probability between 3 and 8 customers: P(3
  3. Average time to serve a customer: 1 / λ = 1 / 12 hours ≈ 0.083 hours (about 5 minutes)
  4. Probability it takes more than 15 minutes (0.25 hours): Using exponential with mean 1/λ = 0.083, probability > 0.25 hours: P(T > 0.25) = exp(-λ * 0.25) where λ = 12 customers/hour
  5. Probability serving takes between 5 and 10 minutes (0.083 to 0.167 hours): P(0.083 0.167) - exp(-λ0.083)

In assessing the accuracy of tax return preparations, answer the following questions:

  1. Calculate P(X ≥ 6): sum P(X=6) + P(X=7) + P(X=8): Actual calculations: 0.2758 + 0.3590 + 0.2044 = 0.8392
  2. Using normal approximation: mean = 65.6, std dev = 3.4363; P(X ≥ 60): with continuity correction P(X > 59.5): Z = (59.5 - 65.6) / 3.4363 = -1.775 probability = 1 - P(Z
  3. Discuss the symmetry of the binomial distribution relative to p=0.5, their approximation to normal when sample size is large, and the benefits of normal approximation.

Paper For Above instruction

The comprehensive understanding of probability distributions and their applications is fundamental in statistical analysis, especially for modeling real-world phenomena. This paper addresses multiple problems involving continuous and discrete distributions, emphasizing their calculations, properties, and applications in various contexts.

Initially, the exploration begins with analyzing four continuous probability distributions: uniform, logistic, normal, and exponential, to compute probabilities for specified events. The uniform distribution over [0,2] has a mean of 1 and variance of 1/3, derived directly from its definition. Geometrical methods or the general cdf form are employed to calculate probabilities such as P(0

The second problem focuses on validating a specific probability density function, f(x) = 3x2 for x in (0,1). The initial step confirms positivity across the support, which holds as 3x2 > 0 for x in (0,1). The integral of f(x) over (0,1) must be one, which is validated through direct integration: ∫01 3x2 dx = 1, satisfying the condition for a legitimate pdf. The expectation E(X) is calculated via ∫01 x * f(x) dx, which results in 0.75. The cumulative distribution function F(x) is derived as F(x) = x3, and this aids in computing the probability P(0.25

The third segment examines traffic fatality statistics through the normal distribution. Using known mean and standard deviation, probabilities are computed via cdf notation, such as P(Crashes

Subsequently, the relationship between the Poisson and exponential distributions is utilized to analyze service times of bank tellers. With a mean of 12 customers per hour, calculations include the probability of serving more than 10 customers, or between 3 and 8, using Poisson formulas. The average service time per customer is the reciprocal of the rate parameter, approximately 0.083 hours (around five minutes). Exponential distribution functions predict the likelihood of service times exceeding 15 minutes or falling within specific minute ranges, illustrating the continuous-time stochastic process.

The final analysis concerns tax return processing competence, modeled with a binomial distribution. Probabilities are summed directly or approximated via the normal distribution; these calculations exemplify how large-sample normal approximations facilitate easier computation where binomial probabilities are cumbersome. The discussion emphasizes the symmetry properties of the binomial distribution and its convergence to normality, highlighting efficiency and accuracy in large samples.

Overall, understanding these foundational probability concepts enables better modeling and decision-making in fields such as finance, engineering, and public policy. The applications ranging from traffic safety analysis to service efficiency and quality control underscore their practical significance, illustrating the power of probability distributions in interpreting data.

References

  • Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.
  • Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
  • Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). Continuous Univariate Distributions. Wiley.
  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage.
  • Mendenhall, W., Beaver, R. J., & Beaver, B. M. (2012). Introduction to Probability and Statistics. Brooks/Cole.
  • Rice, J. A. (2006). Mathematical Statistics and Data Analysis. Duxbury.
  • Newbold, P., Carlson, W., & Thorne, B. (2013). Statistics for Business and Economics. Pearson.
  • Ross, S. M. (2014). Introduction to Probability Models. Academic Press.
  • Agresti, A. (2018). Statistical Thinking: Improving Business Performance. CRC Press.
  • Ostrowski, J. (2019). Applied Probability and Statistics. Springer.

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