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Suppose that a box contains 7 cameras and that 5 of them are defective.
A sample of 2 cameras is selected at random, with replacement. Define the random variable
X as the number of defective cameras in the sample.
(a) Write the probability distribution for X.
(Here there is a table on the left side it says 'k' and on the right side it says 'P(X=k)'.)
There are three values on each side: on the left side there is '0,1,2'
and on the right side there are 3 blank spots. Please find P(x=k).
(b) What is the expected value of X?
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I interpret the problem as if in the experiment the cameras are taken from the set of 7 cameras
one after another with replacement, sequentially.
It is a unique way to take the cameras " with replacement ".
Part (a)
P(X=0) means the probability of having 0 (zero) defective cameras among 2 selected from 7.
P(X=0) = = . <<<---=== every time we select from 7-5 = 2 good cameras
P(X=1) means the probability of having 1 (one) defective camera among 2 selected from 7.
P(X=1) = P(1st is defective, 2nd is good) + P(1st is good,2nd is defective) =
= = = .
P(X=2) means the probability of having 2 (two) defective cameras among 2 selected from 7.
P(X=2) = P(1st is defective, 2nd is defective) =
= = .
Part (a) is complete.
Part (b)
Math expectation = 0*P(X=0) + 1*P(X=1) + 2*P(X=2) = 0 + + = = = 1 = 1 = 1.3.
Part (b) is complete.
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Solved.