The children attending Milena's birthday party are enjoying taking swings at a pinata. Let X= th

(a) $X$ is a geometric random variable with parameter $p=0.2$, since it represents the number of

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The children attending Milena's birthday party are enjoying taking swings at a pinata. Let $X=$ the number of swings it takes Milena to hit the pinata once (since she's the birthday girl, she goes first), and let $Y=$ the number of swings it takes her brother Lucas to hit the pinata once (he goes second). Assume the results of successive swings are independent (the children don't improve, since they're blindfolded), and that each child has a .2 probability of hitting the pinata on any attempt. (a) What is a reasonable probability model for $X ?$ (b) What is a reasonable probability model for $Y ?$ (c) What is a reasonable probability model for $X+Y,$ the total number of swings taken by Milena and Lucas? Explain. (Assume Milena's and Lucas' results are independent.) (d) Use moment-generating functions, along with your answers to (a) and (b), to show that $X+Y$ has a negative binomial distribution. (e) Generalize part (d): If $X_{1}, \ldots, X_{r}$ are independent geometric rvs with common parameter $p$ show that their sum has a negative binomial distribution (f) Does the result of part (e) hold if the probability parameter $p$ is different for each $X_{i}$ (e.g., if Milena has probability .4 on each attempt while access probability is only $\cdot 1 ) ?$

          The children attending Milena's birthday party are enjoying taking swings at a pinata. Let $X=$ the number of swings it takes Milena to hit the pinata once (since she's the birthday girl, she goes first), and let $Y=$ the number of swings it takes her brother Lucas to hit the pinata once (he goes second). Assume the results of successive swings are independent (the children don't improve, since they're blindfolded), and that each child has a .2 probability of hitting the pinata on any attempt.
(a) What is a reasonable probability model for $X ?$
(b) What is a reasonable probability model for $Y ?$
(c) What is a reasonable probability model for $X+Y,$ the total number of swings taken by Milena and Lucas? Explain. (Assume Milena's and Lucas' results are independent.)
(d) Use moment-generating functions, along with your answers to (a) and (b), to show that $X+Y$ has a negative binomial distribution.
(e) Generalize part (d): If $X_{1}, \ldots, X_{r}$ are independent geometric rvs with common parameter $p$ show that their sum has a negative binomial distribution
(f) Does the result of part (e) hold if the probability parameter $p$ is different for each $X_{i}$ (e.g., if Milena has probability .4 on each attempt while access probability is only $\cdot 1 ) ?$

Added by Magdalena M.

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