Since Y is an Exponential distribution with mean β , it’s p.d.f. is given by f ( y) = 1
β
e− y / β , 0 < y < ∞
Let M be the Median of Y. Then we have P[Y ≤ M] = 0.5
M
That is,
∫ f ( y )dy = 0.5
0
That is,
e β ∫
0
1
M
−y/β
dy = 0.5
M
1 e− y / β That is, = 0.5 β −(1/ β ) 0 That is, −e− y / β = 0.5 0 That is, 1 − e − M / β = 0.5 That is, e− M / β = 0.5 That is, − M / β = Log(0.5) That is, M = − β Log(0.5) Thus the median is − β Log(0.5)
M
*
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