In this question, we're told that there is a ballpark. And before a game, vehicles arrived into the ballpark parking lot from the West Side with a rate of 10 per minute and from the east side with a rate of 15 per minute. And we're also told that these arrivals on each side of the parking lot are porcelain processes that air independent from each other. Now, the thing that this question touches on and tries to teach us about is the additive properties of porcelain processes. So if you have two independent person processes, you can characterize them by summing them to create a third process. That is, if they're independent. So for cars entering on the west side of the parking lot, the number of cars that arrive in a given amount of time is a personal random variable, and it has a mean parameter of Lambda Times T or 10 times T and likewise from the east side of the parking lot. The number of cars that arrive in a given amount of time is also a person with mean Lambda Times T or 15 times T. Now we can define a third personal random variable which is the sum of the two first ones since their independence. So we could say that there's some is equal to some third personal random variable. And we can also say that the arrival rate for the third one is equal to the sum of the arrival rates for the first two. So therefore, the total number of arrivals in a given amount of time is also I put some around a variable, but it has a mean of 25 t, and now we can get into the business of answering the questions. So for part, A were asked what the expected number of cars that entered the parking lot in any 10 minute span is, and what is the corresponding standard deviation. So for looking at the total number of cars entering in 10 minutes, we can use this random variable and the mean for put some random variable is equal to Lambda Times T, and in this case, it's Lambda is 25 T is 10 minutes, so we have 250. So 250 is the expected number of cars that enter the parking lot in total in the 10 minute spin now for a Poisson random variable. The variance is also equal to the main, so therefore, the standard deviation of the number of cars that arrive in a 10 minute span is equal to the square root of 250 and that comes out to you 15.81 Yeah, now for Part B were asked that what is the probability that in any particular minute that exactly 12 cars enter from each side? So for this question, it is easier to decompose it back into the two separate possum processes. So the car is entering from the west side and the car is entering from the east side. So we're looking for the probability that number of cars that enter in one minute from the West side is equal to 12, and the number of cars that enter from the East Side in one minute is also equal to 12. And because these are independent personal processes, this probability is equal to the product of the probabilities of each of these events occurring separately. So that's the probability that X one in one minute is equal to 12 times the probability X two in one minute is equal to 12 now, Unlike party, which was a duration of 10 minutes, this is a duration of one minute. So the mean for the first process is 10 times one, and the mean for the second process is 15. So we could say that the probability is equal to you E to the negative 10 times 10 to the exponents, 12 over 12 factorial times e to the negative 15 times 15 to fix opponent 12 over 12 factorial and this comes out to approximately 0.8 So the probability that exactly 12 cars come in from each the west and the east side of the parking lot in a one minute spin is 0.8 Now, for part C, we're asked, what is the probability that exactly 24 cars enter the lot in any particular moment? So this is the total number of cars entering the lot, and we know that the combined rate is 25 t is equal to one minute. So this is a put some random variable with mean lambda three times t. It was 25 and so we're looking for the probability that the total number of arrivals in one minute is equal to 24. This is equal thio e to the negative 25 times 25 to the exponents, 24 over 24 factorial, and this comes out to about zero point 080 So the probability that exactly 24 cars arrive in any given minute is 0.8 And finally, for part D were asked to write an expression for the probability that in any given minute the same number of cars enter enter through both the east and the west sides of the parking lot. So this is just the probability that the number of cars arriving as a result of the first process is equal to the number of the car is arriving as a result of the second process. And keep in mind that we have to plus, um, a random variables here. So the first one has mean so this is one minute, so it has a mean of 10, and the second process has a mean of 15. So this is the probability that both x one and X two are equal. So this is the probability that both X one and X two equals zero, plus the probability that both x one and X two equal one, plus the probability that both x x one and X two equals three and so on. So we can write this as so. It's this summation up to infinity. So there is no upper bound on the number of events that occur as a result of a person process. And the reason why I am multiplying these probabilities is because the two processes are independent processes. Now we can write this using this summation notation. So for X from zero to infinity probability, that number arriving as a result of process one in one minute is equally x times the probability that the number arriving as a result of process to in one minute is also equal to X. And this is equal to the summation of the following. And we can simplify this a little bit. So we have e to the minus 25 and come out of the summation. So this is simplified to 152 the Exponents X and it's expect Auriol squared. And so there may be some fancier ways to simplify this or express it differently. But this is indeed an expression for the probability that in any given minute the same number of cars entered through both sides of the parking lot.