The-number-of-hits-per-minute-on-a-website-is-0-8-It-is-shown-that-the-time-T-between-two-successive-hits-follows-a-negative-exponential-distribution-f-t-0-8e-0-8t-t-0-It-is-required-to-

Question Number 973 by Yugi last updated on 10/May/15

T h e n u m b e r o f h i t s p e r m i n u t e o n a w e b s i t e i s 0.8 . I t i s s h o w n t h a t t h e t i m e T

b e t w e e n t w o s u c c e s s i v e h i t s f o l l o w s a n e g a t i v e e x p o n e n t i a l d i s t r i b u t i o n

f (t) = 0.8 e^{- 0.8 t} (t ⩾ 0) . I t i s r e q u i r e d t o d e t e r m i n e t h e p r o b a b i l i t y t h a t t h e t i m e

b e t w e e n t h e 1 s t h i t a n d t h e 51 s t e x c e e d s o n e h o u r . O n e a p p r o a c h t o t h e

c a l c u l a t i o n s t a t e d t h a t t h e r e q u i r e d p r o b a b i l i t y i s

P (n o . o f h i t s i n 60 m i n s < 50) (*)

C a n a n y o n e e x p l a i n t o m e w h y i t i s t h a t

P (t i m e b e t w e e n 1 s t h i t a n d 51 s t h i t > 60 m i n s) = (*) ?

I s a d l y {d o n}^{'} t u n d e r s t a n d t h e l i n k m a d e b e t w e e n t h e t w o w o r d i n g s o f t h e

s e e m i n g l y e q u a l e v e n t s c o n t a i n e d i n t h e p r o b a b i l i t y e x p r e s s i o n s .

C o u l d a s u f f i c i e n t l y d e t a i l e d e x p l a n a t i o n o f c a l c u l a t i n g (*) b e g i v e n a l s o ?

S o m e s o r t o f a p p r o x i m a t i o n u s i n g t h e n o r m a l d i s t r i b i t i o n i s n e e d e d b u t I^{'} m

u n s u r e o f t h e n e c e s s a r y p r o c e d u r e .

P e r h a p s s o m e o n e c o u l d s u g g e s t w h a t b o o k o r e x a m p a p e r s I m a y u s e t o i m p r o v e

m y c o m p e t e n c e i n n o t i c i n g t h e e q u i v a l e n c e o f e v e n t s i n q u e s t i o n s o f t h i s t y p e .

Answered by prakash jain last updated on 11/May/15

Answer Part I : Equality of events

Let us say 1 st hit occurs at t_{1} and 51 st hit occurs

at t_{51} = t_{1} + x, for time between if t_{51} - t_{1} > 60

then it is clear that only upto 50 hit can occur

between t_{1} and t_{1} + 60 .

The following 2 are equal

1 . time between 1 st hit and 51 st hit > 60

2 . Number of event from t_{1} to t_{1} + 60 ⩽ 50

Now consider the following event :

No of hit in 60 minutes ⩽ 50 .

This event does not mention that the interval

starting from t_{1} is considered . So this is not

the same statement .

However, the given exponential distribution

and poisson probability mass function depend

only on the interval . So the event

^{'} No of hits in 60 mins ⩽ 50^{'} is equal to the event

^{'} time betwen 1 st and 51 st {hit}^{'} only for the

given exponential distribution function .

Commented by Yugi last updated on 11/May/15

I s e e n o w . T h a n k s f o r t h e e x p l a n a t i o n!

Answered by prakash jain last updated on 11/May/15

Answer Part II : Probability of k events in

interval T

If the time between successive event is given

by probability distibution function = λ e^{- λ t}

then the probability function for k events

in interval T is \frac{{(λ T)}^{k} e^{- λ T}}{k!}

In the question you are required to

find probability of upto 50 events . You

can calculate that using the above formula .