Question

The U.S. government has devoted considerable funding to missile defense research over the past 20 years. The latest development is the Space-Based Infrared System (SBIRS), which uses satellite imagery to detect and track missiles (Chance, Summer 2005) The probability that an intruding object (e.g., a missile) will be detected on a flight track by SBIRS is .8. Consider a sample of 20 simulated tracks, each with an intruding object. Let x equal the number of these tracks where SBIRS detects the object.

a. Demonstrate that x is (approximately) a binomial random variable.
b. Give the values of p and n for the binomial distribution. .8.20
c. Find P(x = 15), the probability that SBIRS will detect the object on exactly 15 tracks. .17456
d. Find P(x lessthanorequalto 15), the probability that SBIRS will detect the object on at least 15 tracks. .804208
e. Find E(x) and interpret the result. 16

Answer 1

a) Let the random variable X= "number of these tracks where SBIRS detects the object." in order to use the binomial probability distribution we need to satisfy some conditions:

1) Independence between the trials (satisfied)

2) A value of n fixed , for this case is 20 (satisfied)

3) Probability of success p =0.2 fixed (Satisfied)

So then we have all the conditions and we can assume that:

`X \sim Bin(n =20, p=0.8)`

b)
`X \sim Bin(n =20, p=0.8)`

c)
`P(X=15)=(20C15)(0.8)^(15) (1-0.8)^(20-15)=0.17456`

d)
`P(X \geq 15) = P(X=15)+ .....+P(X=20)`

`P(X=15)=(20C15)(0.8)^(15) (1-0.8)^(20-15)=0.17456`

`P(X=16)=(20C16)(0.8)^(16) (1-0.8)^(20-16)=0.218`

`P(X=17)=(20C17)(0.8)^(17) (1-0.8)^(20-17)=0.205`

`P(X=18)=(20C18)(0.8)^(18) (1-0.8)^(20-18)=0.137`

`P(X=19)=(20C19)(0.8)^(19) (1-0.8)^(20-19)=0.058`

`P(X=20)=(20C20)(0.8)^(20) (1-0.8)^(20-20)=0.012`

`P(X\geq 15)=0.804208`

e)
`E(X) = np = 20*0.8 = 16`

Explanation:

Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

The probability mass function for the Binomial distribution is given as:

`P(X)=(nCx)(p)^x (1-p)^(n-x)`

Where (nCx) means combinatory and it's given by this formula:

`nCx=(n!)/((n-x)! x!)`

Part a

Let the random variable X= "number of these tracks where SBIRS detects the object." in order to use the binomial probability distribution we need to satisfy some conditions:

1) Independence between the trials (satisfied)

2) A value of n fixed , for this case is 20 (satisfied)

3) Probability of success p =0.2 fixed (Satisfied)

So then we have all the conditions and we can assume that:

`X \sim Bin(n =20, p=0.8)`

Part b

`X \sim Bin(n =20, p=0.8)`

Part c

For this case we just need to replace into the mass function and we got:

`P(X=15)=(20C15)(0.8)^(15) (1-0.8)^(20-15)=0.17456`

Part d

For this case we want this probability:
`P(X\geq 15)`

And we can solve this using the complement rule:

`P(X \geq 15) = P(X=15)+ .....+P(X=20)`

`P(X=15)=(20C15)(0.8)^(15) (1-0.8)^(20-15)=0.17456`

`P(X=16)=(20C16)(0.8)^(16) (1-0.8)^(20-16)=0.218`

`P(X=17)=(20C17)(0.8)^(17) (1-0.8)^(20-17)=0.205`

`P(X=18)=(20C18)(0.8)^(18) (1-0.8)^(20-18)=0.137`

`P(X=19)=(20C19)(0.8)^(19) (1-0.8)^(20-19)=0.058`

`P(X=20)=(20C20)(0.8)^(20) (1-0.8)^(20-20)=0.012`

`P(X\geq 15)=0.804208`

Part e

The expected value is given by:

`E(X) = np = 20*0.8 = 16`