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Part BCSIR NET June 2025the-gamma-tail-carries-a-1-plus-x-over-2-factor-the-exponential-does-not

The gamma tail carries a 1 plus x over 2 factor the exponential does not

A mobile manufacturing company uses two brands of batteries for its mobiles. The life (in years) of batteries of Brand I follows an exponential distribution with the probability density function f(x) = e−xe^{-x} if x > 0 and 0 otherwise, and that of Brand II follows a gamma distribution with the probability density function g(x) = (x/4)e−x/2(x/4)e^{-x/2} if x > 0 and 0 otherwise. The company uses the batteries of Brands I and II in proportion of 20% and 80% respectively, in its mobiles. The probability that a randomly selected mobile has the battery life more that 2 years is

  1. A.(13/5)e−2(13/5)e^{-2}
  2. B.(1/5)(e−2(1/5)(e^{-2} + 2e−1)2e^{-1})
  3. C.(1/5)(e−2(1/5)(e^{-2} + 8e−1)8e^{-1})✓ Correct
  4. D.(1/5)(4e−2(1/5)(4e^{-2} + 2e−1)2e^{-1})

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests moments and tails.

See pricing

50 are analysed free — try those first.

The trap it tests

Moments and tails

A moment assumed to exist, or tail behaviour assumed to be tame.

Drill statements like this

Related counterexample: The maximum of independent exponentials is exponential

More on this topic

The chapter behind this: The standard distributions — free to read

From Probability › Standard discrete and continuous distributions

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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