If an individual has never had a previous automobile acciden

If an individual has never had a previous automobile accident, then the probability he or she has an accident in the next h time units is Bh+o(h); on the other hand, if he or she has ever had a previous accidnet, then the probability is ah +o(h). Find the expected number of accidents an individual has by time t.

Solution

Given that

If an individual has never had a previous automobile accident,

then the probability he or she has an accident in the next h time units is Bh+o(h);

if he or she has ever had a previous accidnet, then the probability is ah +o(h).

Let X be a random variable that number of accidents an individual has.

X takes value 0 with probability Bh + o (h)

and take value 1 with probability ah + o(h)

Ant by time t

X takes value 0 with probability Bt + o (t)

and take value 1 with probability at + o(t)

Expectation of X is

E(X) = x*p(x)

= [ 0 * Bt + o (t) ] + [ 1 * at + o(t) ]

= 0 + at + o(t)

E(X) = at + o(t)