Statistics Marathon & Questions (5 Viewers)

[Flop21]

So I replaced x with e^Y. How would I show this is a normal distribution?

 

[InteGrand]

So I replaced x with e^Y. How would I show this is a normal distribution?

 

[Flop21]

Is the pdf of this transformed, Y = ln(X) not equal to the original pdf but with X = e^Y?

 

[InteGrand]

Is the pdf of this transformed, Y = ln(X) not equal to the original pdf but with X = e^Y?

It is not. There is an adjustment factor.

 

[Flop21]

It is not. There is an adjustment factor.

Thanks I've now realised that this is a thing.



Also how would you find P(X + Y) < 1 given fX,Y(x,y).

 

[InteGrand]

Thanks I've now realised that this is a thing.



Also how would you find P(X + Y) < 1 given fX,Y(x,y).

If it's a continuous case (joint PDF), we could integrate the joint PDF over the region in the x-y plane S := {(x,y) : x + y < 1}. (In other words, calculate a double integral.)

 

[Flop21]

Is this correct?

H0 = 0.55 vs H1 = >0.55

[0.6328 - 0.5227]/(0.0873)/sqr(7) = 3.337

P(z > 3.3337) > 0.0001


Thus, there is not enough evidence.

0.6328 - 0.5227

 

[Flop21]

Why does the -1/10 go away when evaluating this?

 

[sida1049]

Why does the -1/10 go away when evaluating this?

Yeah lol, it shouldn't.

 

[Flop21]

If it's a continuous case (joint PDF), we could integrate the joint PDF over the region in the x-y plane S := {(x,y) : x + y < 1}. (In other words, calculate a double integral.)

Sorry it's taking me ages to understand all this.

So for example this one... I get why the integral's limits are for Y, but why is the X limit 0..30, and not 0...30-Y similar to the Y limit???

 

[kawaiipotato]

Sorry it's taking me ages to understand all this.

So for example this one... I get why the integral's limits are for Y, but why is the X limit 0..30, and not 0...30-Y similar to the Y limit???

Draw the line y = 30-x (rearranging is x+y=30) on the cartesian plane. For x,y > 0, this is the first quadrant.
When you want the probability P(Y < 30-X), this is the double integral over the region bounded by the line x+y=30, x=0, y=0.
For double integrals, the outer limits are always constant whilst the inner ones depend on the next variable we integrate with respect to.
The order of integration in that integrand was dydx, so it means we integrate w.r.t y first, and then w.r.t x.
So we want to have bounds on the integral such that it "covers" this whole region (the region I first presented, which will be a triangle in the first quadrant of the plane).
If we restrict it for: 0 less than y less than 30-x and then 0 less than x less than 30 then this will cover this region we want. (see: https://www.desmos.com/calculator/nmfg4p0cw4)
I'm not sure why it keeps cutting off my message randomly.
(For some reason having inequality signs in text deletes the message...)

 

[kawaiipotato]

Try searching up for "double integration and limits" problems online (YouTube) and it should be much clearer

 

[InteGrand]

Draw the line y = 30-x (rearranging is x+y=30) on the cartesian plane. For x,y > 0, this is the first quadrant.
When you want the probability P(Y < 30-X), this is the double integral over the region bounded by the line x+y=30, x=0, y=0.
For double integrals, the outer limits are always constant whilst the inner ones depend on the next variable we integrate with respect to.
The order of integration in that integrand was dydx, so it means we integrate w.r.t y first, and then w.r.t x.
So we want to have bounds on the integral such that it "covers" this whole region (the region I first presented, which will be a triangle in the first quadrant of the plane).
If we restrict it for: 0 less than y less than 30-x and then 0 less than x less than 30 then this will cover this region we want. (see: https://www.desmos.com/calculator/nmfg4p0cw4)
I'm not sure why it keeps cutting off my message randomly.
(For some reason having inequality signs in text deletes the message...)

Not sure if this is the reason but make sure to have spaces between the inequality signs and neighbouring text when typing them on BOS (otherwise it appears to mess up sometimes).