Question

The random variable x is uniform in the interval (0, 1).

Find the density of the random
variable y = -In X

Answer 1

Answer:

X∼U(0,1)

fX(x)=1

To find the distribution of −2logX=Y(say)

First of all let’s settle out domain.

0≤X≤1

Taking log

−∞<logX≤0

Multiplying −2 we have

0≤−2logX<∞

0≤Y<∞

To find the distribution, we begin with Cumulative distribution function of Y

FY(y)=P(Y≤y)=P(−2logX≤y)=P(logX≥y−2)=1−P(logX<y−2)=1−P(X<e−y2)=1−FX(e−y2)

Taking the derivative of both sides with respect to variable y we have:

fY(y)=−fX(e−y2)(e−y2)(−12)

fY(y)=12e−y2y∈[0,∞)

Step-by-step explanation

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