Stat 408 Spring 2014 Homework 5 Due Friday February 28

Stat 408 Spring 2014homework 5 Due Friday February 28 By 300 P

Suppose that the probability of suffering a side effect from a certain flu vaccine is 0.005. If 1000 persons are inoculated, find the approximate probability that a) At most 1 person suffers. (use Poisson approximation) b) 4, 5, or 6 persons suffer.

Suppose that the proportion of genetically modified (GMO) corn in a large shipment is 2%. Suppose 50 kernels are randomly and independently selected for testing. a) Find the probability that exactly 2 of these 50 kernels are GMO corn. b) Use Poisson approximation to find the probability that exactly 2 of these 50 kernels are GMO corn.

Suppose a discrete random variable X has the following probability distribution: P(X=1)=p, P(X=k)= ( ) ! 3ln k k , k=2,3,... (possible values of X are positive integers: 1, 2, 3, … ).

a) Find the value of p that would make this a valid probability distribution. b) Find μX=E(X) by finding the sum of the infinite series. c) Find the moment-generating function of X, MX(t). d) Use MX(t) to find μX=E(X). “Hint”: The answers for (b) and (d) should be the same.

Suppose a discrete random variable X has the following probability distribution: f(k)=P(X=k)=ka, k=2,3,…, where a=Φ−1≈0.618034, where Φ is the golden ratio.

a) Find the moment-generating function of X, MX(t). For which values of t does it exist? b) Find E(X).

Given the moment-generating functions: M(t)= (0.3+0.7 e^t)^5 and M(t)=0.45+0.55 e^t, identify the distribution, compute μ and σ^2, and find P(1 ≤ X ≤ 2).

Given M(t)= t^t e^e^(-0.7t), and M(t)=e^0.3 t, find the distribution and parameters.

Suppose a random variable X has the probability density function f(x)=x/ C, 1

a) Find the value of C so that f(x) is valid. b) Find P(X

Suppose a random variable X has the probability density function f(x)=sin x, 0

a) Find P(X

Suppose X has density f(x)=x e^x, 0

a) Find P(X

For each of the following distributions, calculate P(μ−2σ

13. density f(x)=6x(1−x), 0

14. mass f(x)=x / 2, x=1,2,3,…

Paper For Above instruction

The given homework assignment requires comprehensive statistical computations and theoretical understanding of probability distributions, approximation techniques, and statistical properties. Specifically, it involves calculating probabilities, expectations, moment-generating functions, and understanding the characteristics of various distributions through research and application of formulas.

First, the problem involving the Poisson approximation to binomial and hypergeometric distributions will be addressed, utilizing the Poisson distribution as an approximation tool. The calculations involve determining probabilities for specific cases given the mean rate and sample size. For example, calculating the probability that at most one individual exhibits a side effect from a vaccine leverages the Poisson distribution with a mean λ= np = 1000 x 0.005 = 5; the probability of at most one occurrence is the sum P(X=0)+P(X=1), where P(X=k)= e^(-λ) λ^k / k! for k=0,1. Similarly, for the GMO kernel testing, the Poisson approximation is applicable when n is large, and the proportion is small, which aids in simplifying calculations of rare events, like exactly 2 GMO kernels among the sample.

Next, the assignment addresses defining probability distributions and calculating their properties. For instance, the discrete distribution with probabilities described by a series involving natural logs demands identification of the normalization constant p through the sum of probabilities equaling 1, then calculating the expected value μX through summation, and deriving the moment-generating function MX(t), which allows the computation of moments via derivatives at t=0.

Additionally, the problem involving power-law distributions with the golden ratio explores properties of such distributions, including deriving the moment-generating function and expectation, along with implications of the parameter a= Φ−1. The students are asked to identify distributions through their MGFs, such as the binomial, Poisson, or other distributions, based on the form of their moment-generating functions. These require recognizing the form of the MGFs corresponding to binomial, Poisson, geometric, or other known distributions.

Continuing, the assignment involves evaluating probabilities for continuous distributions such as density functions involving x/ C, sin x, and x e^x, within their respective ranges. This entails integrating to find the normalizing constant, calculating probabilities over intervals, expectations, and medians, demonstrating proficiency with calculus in probability contexts.

Finally, the problem related to the normal approximation to various distributions necessitates computing means and variances and estimating probabilities such as P(μ−2σ

References

  • Papoulis, A., & Pillai, S. U. (2002). Probability, Random Variables, and Stochastic Processes. McGraw-Hill.
  • Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury.
  • Johnson, N. L., Kemp, A. W., & Kotz, S. (2005). Univariate Discrete Distributions. Wiley.
  • Ross, S. M. (2014). Introduction to Probability Models. Academic Press.
  • Hogg, R. V., & Craig, A. T. (2008). Introduction to Mathematical Statistics. Pearson.
  • Resnick, S. I. (1992). Adventures in Stochastic Processes. Birkhäuser.
  • Kreyszig, E. (2011). Advanced Engineering Mathematics. Wiley.
  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Wiley.
  • Wilks, S. S. (1962). Mathematical Statistics. Wiley.

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