Question

This question is based on a Poisson discrete probability distribution. The distribution is important in biology and medicine, and can be dealt with in the same way as any other discrete distribution. Red blood cell deficiency may be determined by examining a specimen of blood under the microscope. The data in Table B gives a hypothetical distribution of numbers of red blood cells in a certain small fixed volume of blood from normal patients. Theoretically, there is no upper limit to the value of a POISSON distribution. In reality, you can force only so many red blood cells into a given volume.  Copy the data from Table B into columns of the EXCEL worksheet, name the columns, and view the table.]

0	0.00000
1	0.00000
2	0.00000
3	0.00001
4	0.00002
5	0.00010
6	0.00031
7	0.00085
8	0.00204
9	0.00435
10	0.00839
11	0.01468
12	0.02355
13	0.03488
14	0.04797
15	0.06157
16	0.07410
17	0.08392
18	0.08977
19	0.09097
20	0.08758
21	0.08030
22	0.07027
23	0.05883
24	0.04720
25	0.03635
26	0.02692
27	0.01920
28	0.01320
29	0.00876
30	0.00562
31	0.00349
32	0.00210
33	0.00123
34	0.00069
35	0.00038
36	0.00020
37	0.00011
38	0.00005
39	0.00003
40	0.00001
41	0.00001
42	0.00000
43	0.00000
44	0.00000
45	0.00000
46	0.00000
47	0.00000
48	0.00000
49	0.00000
50	0.00000
51	0.00000
52	0.00000
53	0.00000
54	0.00000
55	0.00000
56	0.00000
57	0.00000
58	0.00000
59	0.00000
60	0.00000
61	0.00000
62	0.00000
63	0.00000
64	0.00000
65	0.00000
66	0.00000
67	0.00000
68	0.00000
69	0.00000
70	0.00000
71	0.00000
72	0.00000
73	0.00000
74	0.00000
75	0.00000
76	0.00000
77	0.00000
78	0.00000
79	0.00000
80	0.00000
81	0.00000
82	0.00000
83	0.00000
84	0.00000
85	0.00000
86	0.00000
87	0.00000
88	0.00000
89	0.00000
90	0.00000
91	0.00000
92	0.00000
93	0.00000
94	0.00000
95	0.00000
96	0.00000
97	0.00000
98	0.00000
99	0.00000
100	0.00000

8. What is the probability that a blood sample from this distribution will have exactly 20 red blood cells?

9. What is the probability that a blood sample from a normal person will have between 19 and 26 red blood cells? HINT: See questions 3 and 4.

10. What is the probability that a blood sample from a normal person would have fewer than 10 red blood cells?

11. What is the probability that a blood sample from a normal person will have at least 15 red blood cells? HINT: Since there is no theoretical upper limit to the Poisson distribution, the correct way to answer this question is to calculate 1 – probability of fewer than 15 red blood cells. ASSIGNMENT 3 20 INTRODUCTORY STATISTICS LABORATORY

12. A person with a red blood cell count in the lower 2.5 percent of the distribution might be considered as deficient. What is the red blood cell count below which 2.5 percent of the distribution lies? HINT: You need to determine a value X so that if you sum all the probabilities for counts up to and including that value they will sum to at least 0.025. The sum of probabilities of all counts up to but excluding X should be less than 0.025. You can proceed in the following way. Look at the table to guess how many probabilities (P[X = 0] + P[X = 1] + . . ) should be added to give a sum of approximately 0.025. Calculate sums of probabilities for your guess of X. Continue your guessing of X until you get a sum ≥ 0.025 while the sum for X-1 < 0.025.

13. What is the mean red blood cell count in this distribution?

14. What is the variance of red blood cell count in this distribution? HINT: See question 7, and remember it is a Poisson distribution.

15. Is the following statement true (1) or false (0) for this distribution? In a Poisson distribution, the variance is equal to the mean (within rounding error). Record 1 if true, 0 if false.

Answer 1

Let X be the numbers of red blood cells in a certain small fixed volume of blood from randomly chosen normal patients. X has a Poisson distribution given by the probabilities listed in the following table

8. The probability that a blood sample from this distribution will have exactly 20 red blood cells is same as the probability of X=20 (We have defined earlier X as the number of red blood cells in a randomly selected blood sample)

That is we want P(X=20). We will look at the probability corresponding to the row which has x=20 from the above table.

We get P(X=20) = 0.08758

Hence the probability that a blood sample from this distribution will have exactly 20 red blood cells is 0.08758

9) the probability that a blood sample from a normal person will have between 19 and 26 (we will assume here that it is inclusive of both 19 and 26) red blood cells is same as the probability that X is between 19 and 26

That is we want

We get these values from the table and sum them up

The probability that a blood sample from a normal person will have between 19 and 26 red blood cells is 0.49842

10) The probability that a blood sample from a normal person would have fewer than 10 (this means it does not include 10) red blood cells is same as the probability that X is less than 10.

That is we want

The probability that a blood sample from a normal person would have fewer than 10 red blood cells is 0.00768

11) The probability that a blood sample from a normal person will have at least 15 red blood cells is same as the probability that X greater than or equal to 15

That is

But we know that

So using this we get

The probability that a blood sample from a normal person will have at least 15 red blood cells is 0.86285

12. Let X=x be the red blood cell count below which 2.5 percent of the distribution lies. That means we want the value of x for which the probability that the red blood count below x is 0.025.

That means we want

P(X<x) = 0.025

that is we want

P(X=0)+P(X=1)+...+P(X=x-1) = 0.025

We will find the following cumulative distribution of the probabilities

and get theses

We can see that at X=11 the cumulative probability is 0.03075 and at X=10 it is less than 0.025

Hence we can say that the red blood cell count below which 2.5 percent of the distribution lies is 11

13) The mean red blood cell count in this distribution is the expected value of X, which is calculated as

In the excel we create a new column to calculate x*p(X=x) and then sum the column

get these values

the mean red blood cell count in this distribution is 19.2539

14. the variance of red blood cell count in this distribution is calculated as

First we calculate the expectation of using

Then we get the Variance

get these values

The variance of red blood cell count in this distribution is 19.2487

15) We can see that the variance and mean of this distribution are the same.

Hence we can say that

In a Poisson distribution, the variance is equal to the mean ans: True (record 1)