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Q20Section A (Section A)20 of 60

Probability bounds for gamma variable

Let XX be a random variable with the moment generating function M(t)=1(1−4t)5,t<14.M(t)=\frac{1}{(1-4t)^5},\quad t < \frac{1}{4}. Then the lower bounds for P(X<40)P(X < 40), using Chebyshev's inequality and Markov's inequality, respectively, are

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