probability

Suppose that we have three engines, which we turn on at time 0. Each engine will eventually fail, and we model each engine"s lifetime as exponentially distributed with parameter λ. The lifetimes of different engines are independent. One of the engines will fail first, followed by the second, and followed by the last. Let Ti be the time of the first failure·T, be the time of the second failure, and 13 be the time of the third failure. For answers involving algebraic expressions, enter "lambda" for λ and use "exp "for exponentials. Follow standard notation.

1. Determine the PDF of T1. For t>0,

fT1(t)=

2. Let X=T2−T1. Determine the conditional PDF fX|T1(x|t). For x,t>0,

fX|T1(x∣t)=

3. Find E[T3]=

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  1. Can some one help on this

    thanks

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  2. fT1(t)=(3*lambda)*e^(-3*lambda*t)
    fX|T1(x∣t)=(2*lambda)*e^(-2*lambda*x)
    E[T3]=(11)/(6*lambda)

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  3. E[T3] should be this....1/(3*lambda)+1/(2*lambda)+1/lambda

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  4. 1. (3*lambda)*e^(-3*lambda*t)
    2. (2*lambda)*e^(-2*lambda*x)
    3. Yes..They are independent
    4. lambda*e^(-lambda*y)
    5. Yes..They are independent
    6. 3*lambda*e^(-lambda*t)*(1-e^(-lambda*t))^2
    7. 1/(3*lambda)+1/(2*lambda)+1/lambda

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