Inverse Transform Method

[Sunil Chaudhary]

Hi,
Under Inverse Transform Method For Continuous Distributions, the study material mentions sort of proof concluding as:
P[U<=F(x)] = F(x)

Could someone please elaborate on above final conclusion as I am unable to understand the same.

Thanks.
Sunil

 

[Julie Lewis]

The distribution of U is U(0,1). So the CDF of U is F(u) = P(U <= u) = u. Hence P(U <= F(x)) = F(x).

You might find it helpful to consider a numerical example. If U is uniformly distributed over the interval (0,1), then U can take any value between 0 and 1, and each value is equally likely. So, P(U < 0.5) = 0.5, P(U <0.8) = 0.8, etc.

 

[Sunil Chaudhary]

The distribution of U is U(0,1). So the CDF of U is F(u) = P(U <= u) = u. Hence P(U <= F(x)) = F(x).

You might find it helpful to consider a numerical example. If U is uniformly distributed over the interval (0,1), then U can take any value between 0 and 1, and each value is equally likely. So, P(U < 0.5) = 0.5, P(U <0.8) = 0.8, etc.

Thanks Julie.

Sunil.