a) Let X₁, X₂, X3.....X30 be a random sample of size 30 from a population distributed with the following probability density function: f(x)=2 if 0

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a) Let X₁, X₂, X3,...,X30 be a random sample of size 30 from a population distributed with the
following probability density function:
f(x) = 2
e-, if 0<x<∞
otherwise
Suppose that Y = ₁ X₁. Use
i) the moment generating function technique to find the probability distribution function of Y.
Write down the density function of Y.
ii) the Central Limit Theorem to compute P(40 <Y <80).
iii) the Chebychev inequality to find the lower bound of P(40 <Y < 80).
Transcribed Image Text:a) Let X₁, X₂, X3,...,X30 be a random sample of size 30 from a population distributed with the following probability density function: f(x) = 2 e-, if 0<x<∞ otherwise Suppose that Y = ₁ X₁. Use i) the moment generating function technique to find the probability distribution function of Y. Write down the density function of Y. ii) the Central Limit Theorem to compute P(40 <Y <80). iii) the Chebychev inequality to find the lower bound of P(40 <Y < 80).
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