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(Solved): please make the answer as detailed as possible  (a) A random variable \( X \) has a moment gene ...



please make the answer as detailed as possible 

(a) A random variable \( X \) has a moment generating function
\[
M_{X}(t)=e^{2 t+2 t^{2}} .
\]
Write down the mean and varia
(a) A random variable \( X \) has a moment generating function \[ M_{X}(t)=e^{2 t+2 t^{2}} . \] Write down the mean and variance of \( X \). (b) Let \( X \) be an exponential random variable with probability density function: \[ f(x)=\left\{\begin{array}{ll} \lambda e^{-\lambda x} & x \geq 0 \\ 0 & x<0 \end{array}\right. \] i) Find the moment generating function of \( X \) and hence calculate the mean and variance of \( X \). ii) Suppose the time to build a house is an exponential distribution with parameter \( \lambda=1 / 2 \) years. Suppose the cost per year is \( € 100,000 \) with a fixed design cost of \( € 50,000 \). What is the average budget required to build a house? What is the variance of the average budget required to build a house? And what is the probability that a minimum of \( € 200,000 \) is required?


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Given Moment generating function as M(x) = e^(2t+2t^2).. Mean = derivative of moment generating function at t=0.. Derivative of M(x) = (2+4t) ×e^(2t+2
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