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We have provided the parameter of a Poisson random variable, $X .$ For each exercise. a. determine the required probabilities. Round your probability answers to three decimal places. b. find the mean and standard deviation of $X .$ $$\lambda=6.3 ; P(X=7), P(5 \leq X \leq 8), P(X \geq 2)$$

Hello. This is Problem 83. We're working with a pass on distribution. It is called a p. M. It's a probability mass function. So the probability that X is equal to x. Is he good too? Lambda? To the X. E. To the negative lambda over X factorial. We're going to work on part B first. So part B. Were asked what is the mean of this distribution? Well it is going to be lambda and were given the Atlanta 4.7. So the mean equals 4.7. The standard deviation is equal to the square root. What does that mean? Which is lambda? So it's going to be squirt of land which is equal to square it A 4.7. Um And rounded to one decimal place. The answer is 2.2. No for part A. For us what is the probability that X is equal to three? Uh huh. Plugging in the values into this equation we get that it's 4.7 to the third. Multiplied by E. To the negative 4.7 divided by 4.7 pictorial. Um We will be using Excel to actually get the answer. Um Now for the probability that acts is in between five and seven. We need to multiple we need to add through probabilities so we need to add the probably that accesses five pleasant probably at X 06 And the probability that X is equal for seven because this means that we're adding probabilities including five and seven. And in between. Okay. And the last part is the probability that X is greater than two. We're going to use a complementary role since the probability is out of one we're going to do one minus everything. That is not greater than two. Which is a Probably that X is equal to zero um minus the probability that X is equal to one. And the probability that banks is equal to notice that I started with probably of X is equal to zero. Or that is because the domain of the person random variable is um, X is in uh wow, it's could be different. There only whole numbers, right? But this notation means that it's in and it's between zero concludes their own. He goes to infinity. Okay, close to infinity. So now we will use Excel to figure out the numbers. The problem is, yeah, so this is the first one right here, this is the probability that X is equal to three. So, in exile, what it does is we need to say, well, distribution we're working with which is a Poisson distribution. Um this right here is going to be a and uh I mean um it's a 4.7 and we need to put uh falls because we're dealing with the uh P. M. F instead of the we're gonna probably mess with you instead of the community distribution function. Okay. So rounded and we get our answer and rounded to three decimal places. They'll just be 0.1157 Okay, Now, for the second part, which is the probably the X is in between five and seven. Um We're going to um I need to add of the three policies, so the same thing as before. Right? Um We um the exes were here, right? The excess put them into five of the six and seven and the lambdas stay the same. Right? So 4.7 and we included false because we're dealing with a P. M. And the answer is 0.401 Run into three decimal places. Um for the last one, which is the probability that x is greater than two. Uh So there's a problem that of one. That is why we do one mine is uh some distribution, remember we cannot include uh zero, the one and two, so that's how we subtract them, and everything else stays the same, and our answer is 0.8 for eight, rounded to three decimal places.

We will come to new marriage. In the current problem we are given Lambda is equals 2, 3. And we have to find the probabilities for 6, 7 and eight. So for probability of X. Is each of the power minus lambda into lambda. To the power X by X. Factorial. Over here, lambda is three. So it is the power minus three into three. To the power X by ex factory. So for probability of six we have into the power minus three into three. To the power six by six factory. And if we use a calculator, we will arrive at the value 0.05 0409. Following the same steps, we will receive the value of P six to P seven, which is the power minus three into three to the power seven by seven factories. Which again on simplification will become this and P eight is this Now? In when we are doing it on calculator, it is pretty much difficult to each time punching these numbers. Right? So there must be some easier way to do this. So I will show you how can they achieve that? So, there is a recovery in relation off or rather I can say instead of this, let us take be off. A divided by B of X. How much will that be it? The power minus lambda into Landers of our X plus one. Divided by X plus one. Factory in into Now, it will be reciprocal. So X factorial it with the power minus lambda into lambda to the power X. Now, if we see here these two will cancel out X factorial cancelled, get canceled. And only X plus one will be remaining. Right? Because X plus one factorial is nothing but X plus one in two. X factorial. Right ex explain as well and so on. So which is ex factory in. So that's why I only X plus one will be remaining. And here also we will have landed to the power X plus one minus X by X plus one. Which is a cultural under to the power one door. Just simply leave that right X plus one. Now this is nothing but P. Of X plus one by P. X. Is equal to lambda by X plus one. So this is something we need to remember. Okay this recursive relation. You should not forget now once you have this you can exhilarate this. Just hold on. We'll show you how it is getting easy. Okay so now imagine Lambda is three. So let us replace three over here. Now 86 we calculate has ensured that is some 0.050 and Blah Blah. OK once I have P six if I put six over here we have 36 plus 17 and P six. And that's how we can get seven. That means the answer that is there on your calculator screen. This P six into three by seven, They get p. seven and then you get the answer on your screen directly. Zero point that zero to whatever. Okay, now once this answer is there on your calculator screen you can go for P. Eight which will be simply three by eight into this P. seven. That is this value. So I hope how it is going to benefit you. You have been able to understand that and if you have any adults, let me know.

Welcome to enumerate in the current problem, lambda is equals 25 And we have to find the probabilities for the values of x equals 012 dot dot dot dot till six. Okay, so we have to find values P0P even piece everything. Now we know the Poggio PMF is it was the power minus lambda into lambda. To the power X divided by X factorial. Which is it's the power minus five into five. To the power X by five. Factory in. Now, if you want to find 40 you have to Put 20 over here and hopes not by factorial. It is X factorial right and over here to places and for that you have to write each and every term on the calculated. So rather than going that way, we will take another approach. But before that we will just find out P zero. That would be to the power minus five into five. To the power zero by zero factorial. Now this and this boat are one and You to keep our -5 Would be 0.00 6738. Okay. Now following the same procedure we can find the one P two P three but I want to be a little faster in the examination hall so I don't want to do all those calculations. What I will do is I would rather have this recovery in relation remembered. Okay because if you see What will be P one, P 1 will be okay instead of lambda. Okay. Um This is lambda only but for us it will be lambda by one in 20 five x 1 into P zero. Same way P two would be five x 2 into P one. P three would be five x 3. The two before would be five by food before and P five would be five by five. Oh I'm sorry P three P three over here. So this would be before correct? And P six would be five by six. P five, correct? No, you might be wondering here also we are taking a lot of steps. How is it reducing our work Now? Think for this P0, you will write in the step five into 0.006738 Now, Was there already on screen? So all you do is answer into five and you got P one. How Much is People? No, 0.03369. Now, imagine for this is this is our P one. So what will you do now with P even with answer you multiply five divide by two. How is this that Okay, this is on screen now On your calculator screen now. So this will be 0.84 2 to 4. Same way this is a P two now. So with P two if I just multiply five by three I get the next value which will be 0.140374 then 0.175 467 Then 0.175 467. And you think how can these two probabilities equal? We'll look at here Here five x 5 so it will cancel. So whatever is pay for that only will become pre fight. That's how these two are equal. And then the final one is 0.146 2 to 3. So I hope you could understand this. Let me know if you have any questions.

Thanks. Uh, awesome probability for X is equal to three is equal to 6.3 equal. Negative. 6/3 Victoria, which is equal toe point or 89