The number K of customers that arrive at Jay’s supermarket in a given day has the PMF pK(k) = λ^ke^-λ/k! k = 0, 1, 2, . . . Independently of K, the number of items N that any customer purchases from the supermarket has the PMF pN(n) = μ^ne^-μ/n! n = 0, 1, 2, . . . Determine the mean and the

Question

The number [latex]K[/latex] of customers that arrive at Jay’s supermarket in a given day has the PMF

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

Independently of [latex]K[/latex] , the number of items [latex]N[/latex] that any customer purchases from the supermarket has the PMF

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

Determine the mean and the z-transform of the PMF of [latex]\,\displaystyle Y[/latex], the total number of items that the store sells on an arbitrary day.

Accepted Answer

Let [latex]K=k[/latex], and let [latex]N_i[/latex] denote the number of items purchased by customer [latex]i[/latex], [latex]\,\displaystyle 1 \leq i \leq k[/latex]. Then

[latex]\,\displaystyle \begin{aligned} \left.Y ight\vert _{K=k} & =N_1+N_2+\cdots+N_k \ G_{Y \mid K}\left (z \mid k ight ) & =E\left[z^{Y \mid K} ight]=E\left[z^{N_1+N_2+\cdots+N_k} ight]=E\left[z^{N_1} z^{N_2} \cdots z^{N_k} ight] \ & =E\left[z^{N_1} ight] E\left[z^{N_2} ight] \cdots E\left[z^{N_k} ight]=\left[G_N\left (z ight ) ight]^k \ G_Y\left (z ight ) & =\sum_{k=0}^{\infty} G_{Y \mid K}\left (z \mid k ight ) p_K\left (k ight )=\sum_{k=0}^{\infty}\left[G_N\left (z ight ) ight]^k p_K\left (k ight ) \ & =G_K\left (G_N\left (z ight ) ight ) \end{aligned}[/latex]

Since [latex]\,\displaystyle G_K\left (z ight )=e^{\lambda\left (z-1 ight )}[/latex] and [latex]\,\displaystyle G_N\left (z ight )=e^{\mu\left (z-1 ight )}[/latex] , we have that

[latex]\,\displaystyle G_Y\left (z ight )=\exp \left (\lambda\left (e^{\mu\left (z-1 ight )}-1 ight ) ight.[/latex]

Question 7.10: The number K of customers that arrive at Jay’s supermarket i......

Related Answered Questions

Verified Answer:

Verified Answer:

The z-transform of the PMF of a discrete random variable K is given by GK(z) = A[10 + 8z²/2 – z] a. What is the expected value of K? b. Find , the probability that K has the value 1. ...

Verified Answer:

Explain why the function F(z) = z² + z – 1 is or is not a valid z-transform of the PMF of a random variable. ...

Verified Answer:

Is the following function a valid z-transform of a PMF? If so, what is the PMF? g(z) = 1 – a/1 – az 0 < a < 1 ...

Verified Answer:

Determine in an efficient manner the fourth moment of the random variable X whose PDF is given by fX(x) = 32x²e^-4x x ≥ 0 ...

Verified Answer:

A communication channel is degraded beyond use in a random manner. A smart student figured out that the duration Y of the intervals between consecutive periods of degradation has the PDF fY(y) = {0.2^4y³e^-0.2y/3! y ≥ 0 What is the s-transform of the PDF of Y? ...

Verified Answer:

For what value of K is the function A(s) = K/(s + 5) a valid s-transform of a PDF? ...

Verified Answer:

Find the mean and second moment of the random variable whose PDF has the characteristic function ΦX(w) = exp(jwμX – w²σ²X/2) ...

Verified Answer: