ISS PREVIOUS YEAR 2016 PAPER-1 SET-A Q.no.- 12
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ISS PREVIOUS YEAR 2016 PAPER-1 SET-A
![For the distribution:
f(x) = [1 / B(p, q)] × [x^(p − 1) / (1 + x)^(p + q)],where 0 < x < ∞, p > 0, q > 0,
find the harmonic mean.
Options:
(a) p / (p + q) (b) 1 / p (c) (p − 1) / q (d) (p + 1) / (q − 1)](https://static.wixstatic.com/media/8ffd4d_929597f8730a4e7fb508b389d8141e98~mv2.jpg/v1/fill/w_400,h_400,al_c,q_80,enc_avif,quality_auto/8ffd4d_929597f8730a4e7fb508b389d8141e98~mv2.jpg)
Question 12
To compare the lifetimes of bulbs produced by two companies A and B, one bulb from each company was selected at random and their lifetimes were observed.
Assume that the lifetimes follow exponential distributions with:
Mean(A) = 1000 daysMean(B) = 800 days
What is the probability that the bulb from company B fails before the bulb from company A?
Options: (a) 4/9 (b) 5/9 (c) 25/81 (d) 25/41
Solution
Let:
X = lifetime of bulb from company AY = lifetime of bulb from company B
X ~ Exp(λ₁), Y ~ Exp(λ₂)
Step 1: Rate Parameters
Mean = 1 / λ
λ₁ = 1 / 1000λ₂ = 1 / 800
Step 2: Required Probability
P(B fails first) = P(Y < X)
Step 3: Standard Result
P(Y < X) = λ₂ / (λ₁ + λ₂)
Step 4: Substitute Values
Paste this inside Word Equation:
(1/800) / (1/1000 + 1/800)
Step 5: Simplification (Step-by-Step)
1/1000 + 1/800= (4 + 5) / 4000= 9 / 4000
Now:
(1/800) ÷ (9/4000)= (1/800) × (4000/9)= 4000 / 7200= 5 / 9
Final Answer
P(Y < X) = 5/9
Shortcut Formula (Very Important)
Directly write in Word Equation:
P(Y < X) = λ₂ / (λ₁ + λ₂)
ISS PREVIOUS YEAR 2016 PAPER-1 SET-A CLICK HERE TO DOWNLOAD SOLUTION
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