Problem 6.1: Show that if a random variable has a uniform density with the parameters \(\alpha \) and \(\beta \), the probability that it’ll take on a value less that \(\alpha +p(\beta -\alpha )\) is equal to p .

Problem 6.16: If X has an exponential distribution, show that

Problem 6.17: If X is a random variable having an exponential distribution with parameter \(\theta \), find the moment generating function of the random variable \(Y=X-\theta \).

Problem 6.2: Prove Theorem 6.1.

Problem 6.24: If the random variable T is the time of failure of a commercial product and the values of its probability density and distribution function at time t are \(f(t)\) and \(F(t)\), then its failure rate is given by

1. Show that if T has an exponential distribution, the failure rate is constant.

Problem 6.34: If X is a random variable having a normal distribution with the mean \(\mu \) and the standard deviation \(\sigma \), find the moment generating of \(Y=X-c\), where c is a constant, and use it to rework exercise 6.33

Problem 6.37: If X is a random variable having the standard normal distribution, and \(Y={{X}^{2}}\), show that \(\operatorname{cov}\left( X,Y \right)=0\) even though X and Y are evidently not independent.

Problem 6.41: Prove that if X is a random variable having the Poisson distribution with the parameter \(\lambda \), and \(\lambda \to \infty \), then the moment generating function of

converges to the moment generating function of the standard normal distribution

Problem 6.48: If X and Y have a bivariate normal distribution and \(U=X+Y\) and \(V=X-Y\), find an expression for the correlation coefficient of U and V .

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Solution: The downloadable solution consists of 8 pages, 446 words.
Deliverable: Word Document