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# Normal approximation to Poisson watch

1. The number of telephone calls per day follows a Poisson distribution with mean 2

Use an approximate normal distribution to estimate the probability of at least 17 calls in in 8 days.

Newton.
2. X
= Number of calls in 8 days
~ Po(16)

So X ~ N(16, 16) approximately.

P(X >= 17)
~= P(N(16, 16) >= 16.5) . . . . . continuity correction
= 1 - Phi(0.5/4)
= 0.4503
3. I got the same answer as the book arrived at as P(N(0,1)>0.125)
4. Why phi(0.5/4)? Should it not be Phi((1/4)(17.5-16))?
Yes, if we wanted P(X > 17), ie, P(X >= 18). But the question asks for the probability of "at least 17 calls" not "more than 17 calls"
5. (Original post by Newton)
Why does P(X>17)=P(X>=18)?

Newton.
P(X>17) = P(X>=18) for a discrete integer-valued variable
6. X counts the number of calls so it has to be an integer.

P(X > 17)
= P(17 < X < 18) + P(X >= 18)
= 0 + P(X >= 18)
= P(X >= 18)
7. (Original post by Newton)
Ok. Please tell Me if my understanding is then correct.

It is asking for the probability of at least 17 calls i. e. P(X>=17).

By continuity correction this transforms to P(X>=17.5).

But since X must take an integer value we are looking for P(X>=18).

Newton.
No we end up working out

P(N(16,16)>16.5)

because each interval

P( n-1/2 < N(16,16) < n+1/2)

is a good approximation for P(X=n)
8. (Original post by Newton)
So you minus a half?

Newton.
yes - well as appropriate

if we'd been asked what is P(X <= 17) we'd have worked out P(N(16,16)<17.5)
9. (Original post by Newton)
But that is what we are being asked for, hence the 'at least'.

Newton.

no that would be phrased as "at most 17"

just remember

P(X=n) is approximately P(n-1/2 < N(16,16) < n+1/2)

and add up whichever of these you need
10. (Original post by Newton)
I mean P(X>=17).

Newton.
yes so you need to add

P(X=17)+P(X=18)+P(X=19)+...

and by the approx I gave you this is

P(16.5<N<17.5)+P(17.5<N<18.5)+.. .

= P(16.5<N)
11. (Original post by Newton)
tell Me if my understanding is then correct
why Me?
12. (Original post by yazan_l)
why Me?
Obsessive Compulsive Behaviour.

Newton.
13. /Ok. Please tell Me if my understanding is then correct.

It is asking for the probability of at least 17 calls i. e. P(X>=17).

By continuity correction this transforms to P(X>=17.5).

But since X must take an integer value we are looking for P(X>=18).

Newton./

If its P(X>=17) it must include the integer 17 therefore with the continuity correction it becomes P(X>=16.5).

Just remember what integers you have to include!

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