PSTAT 120A 4. (12 points) Two things can set the fire alarm off in your dorm: a

Question

PSTAT 120A 4. (12 points) Two things can set the fire alarm off in your dorm: a fire or your roommate mi crowaving popcorn a bit too long. Suppose that the fire event is independest of the popeorn event. Denote by pe the probability that a fire occurs, and pr the probability that popcorm is microwaved too long. Let Z be the indicator random variable of the fire alarm going Hint: Remember that for some given event A, the indicator random variable la of the event A is defined as 1 if A occurs o if Af oocurs (a) Find the state space of 2. 7 Panacomes (b) Find the probability mass function of Z. ) Compute EZ] 2 E(x) (+ ) . ? "in this . case EC21 P Cb) cofrct Page 5 of S

Explanation / Answer

Answers to problems:

(a). Values taken by the r.v(random variable) Z are {0,1} where Z=1 if the alarm going off and Z=0 not going off.

(b). P(Z=1)=P(alarm going off)=P(F U M)

where F= event that a fire has broken out

M= event that alarm goes off due to 'microwaving too long';

since the alarm can go off when either of the two events occur that's why union of the events is the required probability.

By the theorem of Inclusion Exclusion,

P(F U M)=P(F) + P(M) -P(FM),

from the problem P(F)=pf, P(M)=pm and since F and M are independent ,P(FM)=P(F)*P(M),

so P(FM)=pfpm, and hence

P(Z=1)=pf + pm - pfpm

P(Z=0) =1-P(z=1)

(c) E(Z)=1.P(Z=1) + 0. P(Z=0) = P(Z=1)

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