Suppose customer arrivals at a post office are modeled by a Poisson process N with intensity λ > 0. Let T1 be the time of the first arrival. Let t > 0. Suppose we learn that by time t there has been precisely one arrival, in other words, Nt = 1. What is the distribution of T1 under this new information? In other words, find the conditional probability P(T1 ≤ s|Nt = 1) for all s ≥ 0.

Step-by-step explanation:

We need to find the conditional probability P( T1 < s|N(t)=1 )  for all s ≥ 0

P( time of the first person's arrival < s till time t exactly 1 person has arrived )

= P( time of the first person's arrival < s, till time t exactly 1 person has arrived ) / P(exactly 1 person has arrived till time t )

{ As till time t, we know that exactly 1 person has arrived, thus relevant values of s : 0 < s < t }

P( time of the first person arrival < s, till time t exactly 1 person has arrived ) / P(exactly 1 person has arrived till time t )

= P( exactly 1 person has arrived till time s )/ P(exactly 1 person has arrived till time t )

P(exactly x person has arrived till time t ) ~ Poisson(kt) where k = lambda

Therefore,

P(exactly 1 person has arrived till time s )/ P(exactly 1 person has arrived till time t )

= [ kse-ks/1! ] / [ kte-kt/1! ]

= (s/t)e-k(s-t)

logv3(x+2)=1

rewrite in exponential form:

logb(x) = y is equal to b^y =x

b = 3, y = 1

now solve for x:

3^1 = x+2)

x = 3-2

x = 1

see the picture for the graph.

option a is correct.

step-by-step explanation:

scale factor: it is used to find proportional measurements.

proportions means having the same ratio.

scale factor is the ratio of the model measurement to the actual measurement in a simplest form.

or we can say that the ratio of any two corresponding lengths in two similar geometric figures

if the scale factor is more than 1, then it is an enlargement,

if the scale factor less than 1, then it is a reduction.

given the two geometric figure:

by definition of proportions, we have