Show that if a random variable has an exponential density with the parameter θ, the probability that it will take on a value less than – θ ∙ ln(1 – ρ) is equal to p for 0 ≥ p < 1.
Answer to relevant QuestionsWith reference to Exercise 6.17, using the fact that the moments of Y about the origin are the corresponding moments of X about the mean, find a3 and a4 for the exponential distribution with the parameter θ. In exercise If ...If the random variable T is the time to 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 at time t (see also Exercise 5.24 ...Show that the normal distribution has (a) A relative maximum at x = µ; (b) Inflection points at x = µ – σ and x = µ + σ. Show that if a random variable has a uniform density with the parameters α and β, the rth moment about the mean equals (a) 0 when r is odd; (b) 1 / r + 1 (β – α / 2)r when r is even. Use the results of Exercise 6.4 to find α3 and α4 for the uniform density with the parameters α and β.
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