The probability density function of a certain random variable is:
where takes values in .
Find the probability that .
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This probability is found by computing the CDF and evaluating at 1 0 0 . Given the PDF above, the CDF is:
F X ( x ) = ∫ x λ e − λ x d x = − e − λ x + C .
Since F X ( ∞ ) = 1 , the integration constant C = 1 , and the CDF is thus:
F X ( x ) = 1 − e − λ x .
The probability that x < 1 0 0 is therefore:
F X ( 1 0 0 ) = 1 − e − 1 0 0 λ .
For any reasonable-sized ( O ( 1 ) ) λ , this probability is incredibly close to 1 !