Question

Insulin-Dependent diabetes (IDD) is a common disorder of children. If a specific state has an incidence of 5 cases per 100,000 per year:

a. Suggest a probability distribution and its parameter(s) that can be used to approximate the number of IDD cases per year in the state.

b. What is the probability that the number of cases of IDD is greater than or equal 3 per 100,000 but less than or equal to 7?

Answer 1

a).here, we can use poisson distribution to to approximate the number of IDD cases per year in the state.

let, X =  the number of IDD cases per year in the state.

so, the parameters be:-

n= 100000

incidence rate of IDD per year:-

$\mu= 5/100000 = 0.00005$

so, X ~ P ( 0.00005)

b).the poisson probability is:-

$P(X=x)=\frac{e^{-\mu}\mu^x}{x!}=\frac{e^{-0.00005}0.00005^x}{x!}$

the probability that the number of cases of IDD is greater than or equal 3 per 100,000 but less than or equal to 7 be:-

$P(3\leq X\leq 7)$

$=P(X=3)+P(X=4)+P(X=5)+P(X=6)+P(X=7)$

$=\frac{e^{-0.00005}0.00005^3}{3!}+\frac{e^{-0.00005}0.00005^4}{4!}+\frac{e^{-0.00005}0.00005^5}{5!}+\frac{e^{-0.00005}0.00005^6}{6!}+\frac{e^{-0.00005}0.00005^7}{7!}$

$=e^{-0.00005*}[\frac{0.00005^3}{3!}+\frac{0.00005^4}{4!}+\frac{0.00005^5}{5!}+\frac{0.00005^6}{6!}+\frac{0.00005^7}{7!}]$

$=2.08*10^{-14}$

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