Probability of all runners finishing within 100 minutes

[Addez123]

Homework Statement

The time it takes for a runner to make a lap is a stocastic variable X (in minutes) with density function
$$fx(x) = (125 - x)/450, 95 \leq x \leq 125 $$
Their times are independent of eachother.

What's the probability that all 8 runners finish within 100 minutes?

Relevant Equations

$$Fx(x) = \int fx(x) dx $$
$$Fx(x) = 125/450 * x - x^2/900$$

The chance of everyone finishing should be
$$Fx(100)^8 = (100 * 125/450 - 100^2/900)^8 = (50/3)^8$$

What am I doing wrong?

 

[Orodruin]

It seems you have just taken an arbitrary primitive function for Fx. You need to give a better argument for the value of the integration constant that you have put to zero.

 

[Addez123]

There's no legit reasoning to get a constant.
The density function goes to zero at 125, it doesn't mean everyone have made the lap within 125 minutes.
Just like the density function starts at 30/450 = 0.06 doesn't mean 6% makes it under 95 minutes. The density function outside these bounderies could be anything, its not defined anywhere.

 

[Orodruin]

There's no legit reasoning to get a constant.

This is simply false. You need to think more about what the requirements on the cdf are.

Just like the density function starts at 30/450 = 0.06 doesn't mean 6% makes it under 95 minutes.

Of course not, the pdf is the pdf and the cdf is the cdf.

 

[Orodruin]

The density function goes to zero at 125, it doesn't mean everyone have made the lap within 125 minutes.

This is also false by the way. The pdf is defined to be non-zero only for $95 < x <125$. Obviously, this means that everybody makes it within 125 minutes because the probability of any given runner not making it within 125 minutes is zero.

 

[FactChecker]

Your calculation of Fx(x) is wrong. The density function, fx(x) is zero for x<95. Therefore, your calculation of Fx(x) must start at x=95. (The integral of fx(x) from 95 to 125 is 1, as it must be.)

 

[Addez123]

It's true that the density function sums up to 1 over that integral. Idk why I thought it didn't.
But given that, now I'm trying to find C.

You get different answer depending on your approach.
Fx(125) = 1 => C = -625/36
Fx(95) = 0 => C = -589/36

Shouldn't it give the same value if indeed what was missing was a constant?

 

[FactChecker]

I should have been more clear. The probability of x between 95 and 125 is 1, so Fx(125)-Fx(95) = 1.
You want Prob( x<100 ) = Fx(100)-Fx(95). Then raise that to the eighth power.

 

[Addez123]

Yes that's correct answer. But I still don't understand how I can get different values for C?

 

[FactChecker]

Yes that's correct answer. But I still don't understand how I can get different values for C?

You are not calculating Fx correctly. You are not taking into account that fx is zero below 95. If you do it correctly, the constant from Fx(95) =0 is C=0 and the constant from Fx(125)= 1 is also C=0.