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Assume a Poisson distribution. Find the followingprobabilities.a. Let λ=6.0, find P(X≥2).b. Let λ=0.5, find P(X≤1).c. Let λ=8.0, find P(Xâ‰...

Question

Assume a Poisson distribution. Find the followingprobabilities.a. Let λ=6.0, find P(X≥2).b. Let λ=0.5, find P(X≤1).c. Let λ=8.0, find P(X≤3).

Assume a Poisson distribution. Find the following probabilities. a. Let λ=6.0, find P(X≥2). b. Let λ=0.5, find P(X≤1). c. Let λ=8.0, find P(X≤3).



Answers

Consider a Poisson distribution with $\mu=3$ .
a. Write the appropriate Poisson probability function.
b. Compute $f(2)$ .
c. Compute $f(1)$
d. Compute $P(x \geq 2)$

In this problem, it is given that access of points on distribution with a mean of 0.4 Mean μ is equal to 0.4. We know that new is equal to Lamberti. So Lamberti is equal to 0.4, probability of X is equal to X. For a poison random variable X is the rest to minus Lombardi. Lombardi. Rest two X divided by X factorial. So this is equal to erased too, minus Lambrecht is 0.4. So E th to -0.4 In two, 0.4 raised to X divided by X. Factorial. Yeah, Firstly it is us to find probability of x equal to zero probability of X equal to zero is equal to the rest too -0.4, 0.4 raised two X. But here X is 0.4 raised to zero, divided by zero factorial. This is equal to the rest too -0.4 in two, 0.4 raised to zero is one divided by zero factorial is again one. So this is equal to erase to minus 0.4 in 21 divided by one. That is nearest to -0.4. Es, Tu -0.4 is equal to zero 67 03. So probability x equal to zero is 0.67, Next we are asked to find the probability of X less than or equal to two, Probability of x less than or equal to two is the some of the probabilities X equal to zero as X ranges from zero. We will start from probability of X equal to zero plus Probability of x equal to one plus Probability of x equal to two, probability of X less than or equal to two. Is probability of X equal to zero plus probability of X equal to one. This probability of X equal to two. This is equal to the range to -0.4 In two, 0.4, raised to zero, Divided by zero factorial. This is probability of x equal to zero plus. It is too -0.4 into 0.4, raised to one divided by one factorial. This is probability X equal to one, blessed H two -0.4 In two, 0.4 raised to divided by two factorial. This is probability x equal to two. This is equal to It is 2 -0.4 into 0.4. Restroom zero divided by zero factorial is equal to zero 67 032 plus, It is 2 -0.4 into 0.4. race to one divided by one factorial is zero 46 813 plus, it is two minus 0.4 into 0.4 is 22 divided by two. Factory in is zero point 05 363. So we had to add 0.67032 plus 0.26813 plus 0.5363. This is equal to zero 99 21. So probability x less than or equal to 0.99- one. Now, probability of x equal to four. We have to find the probability of X equal to four, Probability of x equal to four. is It is 2 -0.4 In two, 0.4 raised to eggs. Here, X is four. raised to four, divided by X. Factorial. That is divided by four factorial. This is equal to zero point 00 07 one. Faith Probability of x equal to four is 0.000715. Next we have to find the probability of X equal to eight, probability of X equal to eight, Probability of x equal to eight. This The rest to -0.4 In two, 0.4 raised to X. That is 0.4 raised to eight Divided by x factorial. That is divided by eight factorial. This is equal to it is -0.4-0.4. Is to it divided by eight factorial is equal to what point 090 Into 10, raised to -8, 1.09, 0 into 10 days to -8. This can also be written as zero 1, 0 into 10 days to -7. And this is approximately equal to zero. So Probability of x equal to eight Is approximately equal to zero.

In this problem, it is given that access of points on distribution with a mean of 0.4 Mean μ is equal to 0.4, we know that new is equal to Lamberti. So Lamberti is equal to 0.4, probability of X is equal to X. For a poison random variable X is the rest to minus Lombardi. Lombardi. Rest two X divided by X factorial. So this is equal to the rest Too minus Lambrecht is 0.4. So E th to -0.4 In two, 0.4, raised to X divided by X. Factorial. Yeah, Firstly it is us to find probability of x equal to zero probability of X equal to zero is equal to the rest too -0.4, 0.4 raised two X. But here X is 0.4 raised to zero, divided by zero factorial. This is equal to the rest too -0.4 in two, 0.4 raised to zero is one divided by zero factorial is again one. So this is equal to erase to minus 0.4 in 21 divided by one. That is the rest to -0.4 E. S to minus 0.4 is equal to zero point 67 03. So probability x equal to zero is 0.67, Next we are asked to find the probability of X less than or equal to two Probability of x less than or equal to two is the some of the probabilities X equal to zero as X ranges from zero, we will start from probability of X equal to zero plus Probability of x equal to one plus Probability of x equal to two, probability of X less than or equal to two. Is probability of X equal to zero plus probability of X equal to one. This probability of X equal to two. This is equal to the range to -0.4 In two, 0.4, raised to zero, Divided by zero factorial. This is probability of x equal to zero plus. It is too -0.4 into 0.4. Raised to one divided by one factorial. This is probability X equal to one, blessed H two -0.4 In two, 0.4 raised to divided by two factorial. This is probability x equal to two. This is equal to It is 2 -0.4 into 0.4. Restroom zero divided by zero factorial is equal to zero 67, 032 plus, It is 2 -0.4 into 0.4. race to one divided by one factorial is zero 46 813 plus, it is two minus 0.4 into 0.4 is 22 divided by two. Factory in is zero point 05 363. So we had to add 0.67032 plus 0.26813 plus 0.5363. This is equal to zero 99 21. So probability x less than or equal to 0.99- one. Now, probability of x equal to four. We have to find the probability of X equal to four, Probability of x equal to four. is It is 2 -0.4 In two, 0.4 raised to eggs. Here, X is 40.4, raised to four, divided by X. Factorial. That is divided by four factorial. This is equal to zero point 00 07 one. Faith Probability of x equal to four is 0.000715. Next we have to find the probability of X equal to eight, probability of X equal to eight, Probability of x equal to eight. This The rest to -0.4 In two, 0.4 raised to X. That is 0.4, raised to eight, Divided by x factorial. That is divided by eight factorial. This is equal to it is -0.4-0.4 is to it divided by eight factorial is equal to what point 090 Into 10, raised to -8. 1.09, 0 into 10 days to -8. This can also be written as zero 1, 0 into 10 days to -7. And this is approximately equal to zero. So Probability of x equal to eight Is approximately equal to zero.

Seven. A possum probability for X is equal to is equal to 1.5 or two if or negative 1.5 over to Victoria, which is 4.2510

In this problem, it is given that X has a poison distribution with mean food. That this mean μ is equal to four. We know that mu is equal to Lamberti, So Lamberti is equal to four. For poison variable X. We have probability of X is equal to X Is the rest to- Lombardi. Lombardi, raised two x divided by X Factorial. The x goes from 0 1 up to infinity, so probability of X is equal to X. Yes, it is too minus. We have landlady is equal to fall, So it is 2 -4 Into Forest two. X divided by ex victorian. Here. First we have to find probability of X is equal to zero, probability horse, X is equal to zero, This is equal to the rest to -4. four raised 2, zero. As here X is equal to zero, so four is 20. Do I get by zero factorial? This is equal to The rest to -4. Four is 20 is one, and zero factorial is also one, so it is two minus four, Which is equal to zero 01 83. So probability of X is equal to 00.0183. Next we have to find the probability of X less than or equal to two, Probability of x less than or equal to two. This is equal to The value of x starts from zero, So probability of X is equal to zero Plus, the probability of X is equal to one. This probability of X is equal to two. As we have to find probability of X less than or equal to two. This will be equal to the probability of X is equal to zero, plus probability of X is equal to one. This probability of X is equal to two. This is equal to it is to -4. four raised 20. For the probability that physical 20, This will be forced to zero divided by zero factorial less For probability of X is equal to one. This will be the rest to -4. four is 2. one divided by one. Factory in plus. For a probability of X is equal to two, this will be the rest to -4. Forest to do divided by two factorial, This is equal to, it is two minus 44 is 20, divided by zero. Factorial is equal to 0.183. Yes, it is two minus four forties to one divided by one. Factorial is equal to 0.7 33 plus It is true -4. Forest to two, divided by two factorial is equal to zero 14 six fights 0.183 plus 0.733 plus 0.1465 is equal to zero point 43 81. So probability of x less than or equal to two is 0.2381 Next we have to find probability of X is equal to four, Probability of X is equal to four is equal to He rest to -4. Four is two eggs. But here X is four, so four is 24 divided by X factorial. That is four factorial, It is 2 -4, multiplied by four is 24 divided by four. Factorial is equal to zero point 19 five ft. So probability of X is equal to four, Next we have to find the probability of X is equal to eight, probability of X is equal to add, this is equal to It is 2 -4, four raised two X. But here X is eight, so four is 28 Divided by X. Factorial. That is divided by eight factorial, it is two minus four, multiplied by four is 28, divided by eight. Factorial is equal to zero point 02 98, So probability of X is equal to eight is 0.029, eight. Thus we have found all the required probabilities.


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