At Least ???

Refer study material for: CT3-06 / IFE: 2008 Examination


Q 6.27 , PAGE 32 ; (Solution 6.27 , page 55)

Atleast five calls is calculated as ... P(X >= 5). Which means greater than 5 including 5.



Q 6.28 (ii), page 35 ; (Solution 6.28 (ii) , page 56)

Atleast fifteen minutes is calculated as ... P(X > 15). Which means greater than 15 & not including 15 !!!


Can anyone explain why the sign is different ( >= eq & greater than) & ( > only greater than) ? In both the cases we are calculating at least values only.

 

Is the second variable (in minutes) continuous (as opposed to discrete)?

If so, this may explain it as generally equal signs are omitted when dealing with continuous variables (because there is zero probability of being 15 EXACTLY).

 

At least

First one is a Poisson dist.

Second one is Exponential dist.

 

OK

X~poisson = Discrete X. can take only whole number values.

Y~Exponential = Continuous. Y takes any real number value


With continuous variable the probability of being exactly any one number is zero. ie probability (y = 15.000000(umpteen zeros) ) is zero (or as good as being zero) so the equal sign is usually omitted as it makes no difference

That doesn't mean to say that you can't include the "equal" bit yourself.

The above may seem confusing. we can get probability y= 15 to some degree of accuracy eg p(14.5<Y<15.5) or p(14.9999995<Y<15.0000005)
but you will notice that the probability gets smaller and smaller as you increase the accuracy until you have a probability of zero at EXACTLY 15.

You can also think of what the difference would be is you had valued P(Y<15) or P(Y<=15). you should arrive at there being no difference.

Hope that makes sense now.

 

To make life easier :

For discrete: at least 5, means P( X>=5 )

For continuous: at least 5, means P( X>5 )

???

 

Yes.

P(X=k) = 0 for a continuous RV