Let X₁, X₂ be IID with \Exp(1), the standard exponential distribution. Show that Z = X₁/X₂ has an F-distribution. Determine the degrees of freedom of this F- distribution.
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- The life of an electronic device is known to have the exponential distributionwith parameter lamda=1/1000 .(i) What is the probability that the device lasts more than 1000 hours?(ii) What is the probability it will last less than 1200 hours?(iii) Find the mean and variance of the life of the electronic device.Express the Binomial distribution as an exponential family distribution. Also express the Beta distribution is an exponential family distribution. Show that the product of the Beta and the Binomial distribution is also a member of the exponential family.Let the life span of a material be given by the probability function f(x)=c²xe^(-cx) (x≥ 0). Determine the average lifetime of this material.
- The annual rainfall (in inches) in a certain region is normally distributed with meanµ = 40 and variance σ^2 = 25. Find the probability that between 35 and 50 inches of rain will fall in theregion in the year 2021. Express your final answer in terms of the standard normalcumulative distribution function Φ. What is the probability that in at least 2 of the next 10 years less than 20inches of rain are recorded in the region?Which distribution's height begins low, increases until the middle, and then decreases such that it has a steep side and a shallow sideIf X ~B(190,0.34), find mean and variance of x. (Rounded in 3dp.)
- (3) Let X = b >(8,-) find E(5+6x) and distribution function.Suppose that the time-to-failure of a system has a distribution with the following pdf: f(x) = 0.1e^(-0.1x) What is the probability that the failure occurs between 5 and 20? a). 0.23 b). 0.47 c). 0.088 d). 0Let Y > 0 be a continuous random variable representing time from regimen start to bone-marrow transplant. Everyone does not survive long enough to get the transplant. Let X > 0 be a continuous random variable representing time from regimen start to death. We can assume X ⊥ Y and model time to death as X ∼ Exp(rate = θ) and time to transplant as Y ∼ Exp(rate = µ). Where Exp(rate = λ) denotes the exponential distribution with density f(z | λ) = λe−λz for z > 0 and 0 elsewhere - with λ > 0. Find the probability that the patient would die before receiving transplant.