Question

Solve for the following probabilities (ranges of X values):
a.       P(X ≤ 7) when m = 15
b.      P(9 ≤ X ≤ 18) when m = 15
c.       P(X ≥ 15) when m = 15
d.      P(12 ≤ X < 20) when m = 15

(1) Solve the probabilities on the PROB worksheet. (To four decimal places) The P(X ≤7) when the poisson mean = 15 is ________.Assume a poisson distribution

(2) Solve the probabilities on the PROB worksheet. (To four decimal places) The P(9 ≤ X ≤ 18) when the poisson mean = 15 is ________.Assume a poisson distribution

(3) Solve the probabilities on the PROB worksheet. (To four decimal places) The P(X ≥ 15) when the poisson mean = 15 is________.Assume a poisson distribution

(4) Solve the probabilities on the PROB worksheet. (To four decimal places) The P(12 ≤ X < 20) when the poisson mean = 15 is ________.Assume a poisson distribution

(5) Which of the following Excel formulas can calculate P(10 ≤ X ≤ 20) when the poisson mean = 15 using the CUMULATIVE Distribution function (F(x))?

=poisson(20,15,1)-poisson(9,15,1)

=poisson(21,15,0)-poisson(10,15,0)

=poisson(21,15,1)-poisson(10,15,1)

=poisson(20,15,0)-poisson(9,15,0)

(6) Which of the following Excel formulas can calculate P(X ≥ 13) when the poisson mean = 15 using the PROBABILITY function (p(x))?

=sum(p(100):p(13)), where p(x) = poisson(x,15,0)

=1 - sum(p(13):p(100)), where p(x) = poisson(x,15,0)

=1 - sum(p(0):p(13)), where p(x) = poisson(x,15,0)

=1 - sum(p(0):p(12)), where p(x) = poisson(x,15,0)

Answer 1

The poisson probability is given as:

$P(X=x)=\frac{e^{-m}m^x}{x!}, x=0,1,2,.....\infty$

1)

$P(X\leq 7)=P(X=0)&plus;P(X=1)&plus;....&plus;P(X=7)$

P(X=0)	3.05902E-07
P(X=1)	4.58853E-06
P(X=2)	3.4414E-05
P(X=3)	0.00017207
P(X=4)	0.000645263
P(X=5)	0.001935788
P(X=6)	0.00483947
P(X=7)	0.010370294
Total=	0.018002193

$P(X\leq 7)=0.0180$

Excel formula used :
Poisson(x,15,0) where x =0,1,.....7
and then summing all the probabilities
Alternate formula:
Poisson(7,15,1) will give the direct result

2)

$P(9\leq X\leq 18)=P(X=9)&plus;P(X=10)&plus;.........P(X=18)$

P(X=9)	0.032407
P(X=10)	0.048611
P(X=11)	0.066287
P(X=12)	0.082859
P(X=13)	0.095607
P(X=14)	0.102436
P(X=15)	0.102436
P(X=16)	0.096034
P(X=17)	0.084736
P(X=18)	0.070613
Total	0.782025

$P(9\leq X\leq 18)=0.7820$

Excel formula used :
Poisson(x,15,0) where x =9,10,.....18
and then summing all the probabilities
Alternate formula:
Poisson(18,15,1)-Poisson(8,15,1) will give the direct result.

3)

$P( X\geq 15)=1-P(X<15)=1-P(X\leq14)\\ \\=1-[P(X=0)&plus;P(X=1)&plus;...&plus;P(X=14)]$

P(X=0)	3.06E-07
P(X=1)	4.59E-06
P(X=2)	3.44E-05
P(X=3)	0.000172
P(X=4)	0.000645
P(X=5)	0.001936
P(X=6)	0.004839
P(X=7)	0.01037
P(X=8)	0.019444
P(X=9)	0.032407
P(X=10)	0.048611
P(X=11)	0.066287
P(X=12)	0.082859
P(X=13)	0.095607
P(X=14)	0.102436
Total=	0.465654

$P( X\geq 15)=1-P(X<15)=1-P(X\leq14)=1-0.4657=0.5343$

Excel formula used to calculate probability upto 14 :
Poisson(x,15,0) where x =0,1,.....14
and then summing all the probabilities and subtracting from 1

Alternate formula:
1-Poisson(14,15,1) will give the direct result

4)

$P(12 \leq X\leq20)=P(X=12)&plus;P(X=13)&plus;....&plus;P(X=20)$

12	P(X=12)	0.082859
13	P(X=13)	0.095607
14	P(X=14)	0.102436
15	P(X=15)	0.102436
16	P(X=16)	0.096034
17	P(X=17)	0.084736
18	P(X=18)	0.070613
19	P(X=19)	0.055747
20	P(X=20)	0.04181
	Total	0.732277

$P(12 \leq X\leq20)=0.7323$

Excel formula used :
Poisson(x,15,0) where x =12,13,.....20
and then summing all the probabilities
Alternate formula:
Poisson(20,15,1)-Poisson(11,15,1) will give the direct result.

(5) Which of the following Excel formulas can calculate P(10 ≤ X ≤ 20) when the poisson mean = 15 using the CUMULATIVE Distribution function (F(x))?
=poisson(20,15,1)-poisson(9,15,1)

(6) Which of the following Excel formulas can calculate P(X ≥ 13) when the poisson mean = 15 using the PROBABILITY function (p(x))?

=1 - sum(p(0):p(12)), where p(x) = poisson(x,15,0)