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2, A random variable has Poisson distribution Poi( a ) Estimate the parameter using the samplex1 = 52, Xz = 48, X3 = 49, X4 =49 Xs = 52, X6 = 50, X, =47, Xa = 48,...

Question

2, A random variable has Poisson distribution Poi( a ) Estimate the parameter using the samplex1 = 52, Xz = 48, X3 = 49, X4 =49 Xs = 52, X6 = 50, X, =47, Xa = 48,

2, A random variable has Poisson distribution Poi( a ) Estimate the parameter using the sample x1 = 52, Xz = 48, X3 = 49, X4 =49 Xs = 52, X6 = 50, X, =47, Xa = 48,



Answers

If the random variable $X$ has a Poisson distribution such that $P(X=1)=$ $P(X=2)$, find $P(X=4)$.

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.

Random variable X follows a solemn process with lambda equals 0.2 and T equals 50. We want to compute a through E. Given this information alone to start off with. As I've already done in the upper left, following the asterisks, we're going to compute the mu or rather the mean of this person process, mu equals lambda, T equals one. And then given that mu we're going to use the Poisson probability formula P of X equals mu to the X over X factorial. Either the negative you to answer these questions. So for instance, for part A Kiev to plug in X equals to U equals one and 2% probability formula one squared over two factorial times. Even the negative first equals 20.184 Next probably the X is less than two is simply the sum of P 0 51. We plug in X equals zero. Next equals one. To obtain even negative 1/2 or again .184. Next in part C, probably X is greater than equals two is simply the complement of PART B or one minus. Probably access less than two plugging in. Our answer from Part B gives us 20.816 Next for part D. The probability that one is less than equal to act is nothing new for the three. So we follow similar steps to be where this is simply P one plus P two plus P three plugging in. You take the the negative first times one plus one half +16 which is equal 2.613 And finally, in part you want to calculate mu and sigma me was one as we've already found single and the standard deviation of you for a sense of sigma equals one as well.

From We're told that we have a Poisson distribution or land is april to 6.3. So since we know it's a Poisson distribution, this means that the probability of why was able to search 0.3 to the y. Either the negative 6.3 over why fact or now here on a We want to find some probabilities here. The first thing we want to find the probability that X. I called it why they called that same thing probability access set until pointing in seven there this would be 6.3 to the seventh. Eat of the negative 6.3 over seven factory. And so you can just type the straight in and when we do this will give us 0.143 five. Now we also want to find probability, okay, that is between five and I. This is the probability of five. What's the probability of suits? What's the probability of seven plus? The probability of a. Mhm. So now we put in all these values, we have 6.3 to the fifth. You know the name of 6.3 over five Electoral. That's probability of five. All 6.3 of the search. Either the nature of 6.3 over six factorial. All 6.3 to the seventh. Each of the native 6.3 over seven factorial. Mhm. Well 6.3 to 8 you know the name of 6.3 over a factory. That's the probability of eight. And so now we just type all this in straight on your calculator. And when we do this should give us 0.5679 And then you just type all this straight in exactly right order. That should give us this fight. Mhm. Now we also want the probability the next is greater than or equal to two. Now this is one minus the probability that x is less than to just using our compliment compliment rule, which is one minus the probability X zero plus the probability access one which is one minus 6.3 to 0 each of the knee of 6.3 over zero factorial or 6.3 to the one, you know the name of 6.3 over one factory. So now when we evaluate this and you just type the straight in on your twitter, this will give us 0.9866 So that's the probability X is right. And in april two mature, remember take the one minus now if we want to find the expected value. Yeah. And the standard deviation expected value for a Poisson distribution is just equal to lambda. And here we are told by up to 6.3 and so they expected value is 6.3. The standard deviation is always equal to the square root of the variance. What we saw in the variance has always land us. This is the square root of 6.3, and then the square root of 6.3 is approximately 2.5 what?

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.


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