Answer:
The variable x will follow a geometric distribution, with mean = 3 and standard deviation = 3.46.
The values for P(x) are:
[tex]P(0)=0.25\\\\P(1)=0.1875\\\\P(2)=0.1406\\\\P(3)=0.1055\\\\P(4)=0.0791\\\\P(5)=0.0593\\\\P(6)=0.0445\\\\P(7)=0.0334\\\\P(8)=0.025\\\\P(9)=0.0188\\\\P(10)=0.0141\\\\[/tex]
Step-by-step explanation:
This kind of random variables can be descripted by the geometrical distribution.
This distribution shows the probability of having an amount of "failures" before the first "success" (or the other way).
Let x be the number of trustworthy FBI agents tested until someone fails the test. The probability of failing a test is p=0.25
Then, the probability of x is:
[tex]P(x)=q^x\cdot p[/tex]
The values for the first x are:
[tex]P(0)=0.75^0\cdot 0.25=0.25\\\\P(1)=0.75^1\cdot 0.25=0.1875\\\\P(2)=0.75^2\cdot 0.25=0.1406\\\\P(3)=0.75^3\cdot 0.25=0.1055\\\\P(4)=0.75^4\cdot 0.25=0.0791\\\\P(5)=0.75^5\cdot 0.25=0.0593\\\\P(6)=0.75^6\cdot 0.25=0.0445\\\\P(7)=0.75^7\cdot 0.25=0.0334\\\\P(8)=0.75^8\cdot 0.25=0.025\\\\P(9)=0.75^9\cdot 0.25=0.0188\\\\P(10)=0.75^{10}\cdot 0.25=0.0141\\\\[/tex]
This probability will be descending as the the variable x increases.
The mean is:
[tex]\mu=\frac{1-p}{p}= \frac{1-0.25}{0.25}=\frac{0.75}{0.25}=3[/tex]
The standard deviation is:
[tex]\sigma=\sqrt{\frac{1-p}{p^2}} =\sqrt{ \frac{0.75}{0.25^2} }=\sqrt{12}= 3.46[/tex]
