Question:

Solution:

I have 3 questions about this solution:

-Where is the uniform distribution over (0, 1) used?

-Why is it P(W>w) as opposed to P(W<w)?

-Why is P(X>w)=P(Y>w)=P(Z>w)=(1-w)?

After being unable to solve the question and then re-reading the solution, I tried P(W<w) instead where I assumed P(X<w)=P(Y<w)=P(Z<w)=w which gave an incorrect answer, but I'm unsure as to why so I'd appreciate it if somebody could help me answer these 3 questions.

*The random variables X, Y and Z are independent and have an identical*

distribution that is uniform over (0, 1). Calculate the median of the

smallest of the three.distribution that is uniform over (0, 1). Calculate the median of the

smallest of the three.

Solution:

*Let W = min(X, Y,Z). Then*

Pr(W > w) = Pr(min(X, Y,Z) > w)

= Pr(X > w, Y > w,Z > w)

= Pr(X > w)Pr(Y > w)Pr(Z > w) (independence)

= (1 − w)^3 = 1/2

when w = 1 − (1/2)1/3 = 0.206.

Pr(W > w) = Pr(min(X, Y,Z) > w)

= Pr(X > w, Y > w,Z > w)

= Pr(X > w)Pr(Y > w)Pr(Z > w) (independence)

= (1 − w)^3 = 1/2

when w = 1 − (1/2)1/3 = 0.206.

I have 3 questions about this solution:

-Where is the uniform distribution over (0, 1) used?

-Why is it P(W>w) as opposed to P(W<w)?

-Why is P(X>w)=P(Y>w)=P(Z>w)=(1-w)?

After being unable to solve the question and then re-reading the solution, I tried P(W<w) instead where I assumed P(X<w)=P(Y<w)=P(Z<w)=w which gave an incorrect answer, but I'm unsure as to why so I'd appreciate it if somebody could help me answer these 3 questions.

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