Question

In the problem, make sure that you are clearly defining random variables, stating their distributions, and writing down the formulas that you are using. (That is, write down the pmf, write down mean and variance formulas.)

The number of particles emitted by a radioactive source over the course of an hour is generally well modeled by a Poisson distribution. (We will see some solid justification of this later on when we discuss the gamma and exponential distributions.) Suppose that the average number of particles emitted by the source in an hour is four.

a. What is the pmf of X=the number of particles emitted by the source in a given hour?

b. What is the probability that the number of particles emitted in the given hour is at least 6?

c. What is the probability that the number of emitted particles in a given hour will be at most 3?

d. What is the probability no particles at all are emitted in a 24 hour period? Leave your answer as a simplified expression. (Hint: Add some random variables together)

Answer 1

(a) λ = 4; P(x) = (e^-λ)(λ^x) / x!

P(x) = (e^-4)(4^x) / x!

(b) By substituting different values for x in the above formula, we can find P(x)

x	P(x) = (e^-4)(4^x) / x!
0	0.0183
1	0.0733
2	0.1465
3	0.1954
4	0.1954
5	0.1563
6	0.1042
7	0.0595
8	0.0298
9	0.0132
10	0.0053
11	0.0019
12	0.0006
13	0.0002
14	0.0001
15	0.0000
16	0.0000

P(x ≥ 6) = 1 - P(x < 6) = 1 - [P(0) + P(1) + P(2) + P(3) + P(4) + P(5)] = 0.2149

(c) P(x ≤ 3) = P(0) + P(1) + P(2) + P(3) = 0.4335

(d) Here, λ = 4 * 24 = 96

P(x) = (e^-96)(96^x) / x!

P(0) = (e^-96)(96^0) / 0! = e^-96 (Almost 0)

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