Books are packed into cartons. The weight W of a book is a continuous random variable with PDF fW(w) = λe^-λw w ≥ 0 The number K of books in any carton is a random variable with the PMF pK(k) = μ^k/k! e^-μ k = 0, 1, 2, . . . If we randomly select a carton and its weight is X, determine a. the

Question

Books are packed into cartons. The weight [latex]W[/latex] of a book is a continuous random variable with PDF

[latex]\,\displaystyle f_W\left (w\right )=\lambda e^{-\lambda w} \quad w \geq 0[/latex]

The number [latex]K[/latex] of books in any carton is a random variable with the PMF

[latex]\,\displaystyle p_K\left (k\right )=\frac{\mu^k}{k !} e^{-\mu} \quad k=0,1,2, \ldots[/latex]

If we randomly select a carton and its weight is [latex]\,\displaystyle X[/latex], determine

a. the s-transform of the PDF of [latex]X[/latex].

b. [latex]E[X][/latex].

c. the variance of [latex]X[/latex].

Accepted Answer

a. The s-transform of the PDF of [latex]W[/latex] is given by

[latex]\,\displaystyle M_W\left (s\right )=\frac{\lambda}{s+\lambda}[/latex]

Similarly, the z-transform of the PMF of [latex]K[/latex] is given by

[latex]\,\displaystyle G_K\left (z\right )=e^{\mu\left (z-1\right )}=\exp \left (\mu\left (z-1\right )\right )[/latex]

Thus, the s-transform of the PDF if [latex]X[/latex] is given by

[latex]\,\displaystyle M_X\left (s\right )=G_K\left (M_W\left (s\right )\right )=\exp \left (\mu\left\{\frac{\lambda}{s+\lambda}-1\right\}\right )=\exp \left (-\frac{\mu s}{s+\lambda}\right )[/latex]

b. The expected weight of the randomly selected carton is

[latex]\,\displaystyle E[X]=E[K] E[W]=\mu\left (\frac{1}{\lambda}\right )=\frac{\mu}{\lambda}[/latex]

c. The variance of [latex]X[/latex] is given by

[latex]\,\displaystyle \sigma_X^2=E[K] \sigma_W^2+\{E[W]\}^2 \sigma_K^2=\mu\left (\frac{1}{\lambda^2}\right )+\left (\frac{1}{\lambda^2}\right ) \mu=\frac{2 \mu}{\lambda^2}[/latex]

Question 7.8: Books are packed into cartons. The weight W of a book is a c......

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