9. A discrete random variable X has a Poisson distribution with mean 6.8. (1)Fin

Question

9. A discrete random variable X has a Poisson distribution with mean 6.8.

(1)Find P(X <5). (2)Find P(X >10). (3)Find P(X <12). (4)Find P(7 X 11).

10. Let Z be a random variable with standard normal distribution. Find a number z0 (1) P(Z>z0)=0.7389
(2) P(Z<z0)=0.0032
(3)P(z0 <Z<z0)=0.6046

so that:

(4)P(z0 <Z<1.47)=0.9126

11. A random variable X has a normal distribution with mean 7 and standard deviation 2.

(1) Find the probability that X is between 5 and 8.

(2) Find the probability that X is greater than 6.

(3) Find a number x0 so that P(X > x0 )=0.8508.

Explanation / Answer

9.

a)

Note that P(fewer than x) = P(at most x - 1).          
          
Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    6.8      
          
x = our critical value of successes =    5      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   4   ) =    0.192030874
          
Which is also          
          
P(fewer than   5   ) =    0.192030874 [ANSWER]

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b)

Note that P(more than x) = 1 - P(at most x).          
          
Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    6.8      
          
x = our critical value of successes =    10      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   10   ) =    0.915066095
          
Thus, the probability of at least   11   successes is  
          
P(more than   10   ) =    0.084933905 [ANSWER]

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c)

Note that P(fewer than x) = P(at most x - 1).          
          
Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    6.8      
          
x = our critical value of successes =    12      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   11   ) =    0.955174911
          
Which is also          
          
P(fewer than   12   ) =    0.955174911 [ANSWER]

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d)

Note that P(between x1 and x2) = P(at most x2) - P(at most x1 - 1)          
          
Here,          
          
x1 =    7      
x2 =    11      
          
Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    6.8      
          
          
Then          
          
P(at most    6   ) =    0.47991622
P(at most    11   ) =    0.955174911
          
Thus,          
          
P(between x1 and x2) =    0.47525869   [ANSWER]  

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