Question

1. Two transmitters send messages through bursts of radio signals to an antenna. During each time slot, each transmitter sends a message with probability p. Simultaneous transmissions result in a loss of the messages. Let X denote the number of time slots until the first message gets through. (a) Describe the underlying sample space 2 of this random experiment and specify a o-field and the probabilities of the underlying events. (b) Show the mapping from 1 to 12x, the range of X. (c) Find the probabilities for the various values of X.

Answer 1

Two transmitters send messages through bursts of radio signals to an antenna. Let T be the signal send at a time slot by 1st transmitter. Let T be the signal send at a time slot by 2nd transmitter. Each transmitter sends a message with probability p. P(17)=P(Iz) = p Let the random variable X represents the number of time slots until the first message gets through. A signal is being received if either of the transmitters sends the message. (IT; ) (T_T) And a signal is lost if both the transmitters send messages simultaneously. T12 a) The objective is to find the sample space S of this random experiment and specify the probabilities of its elementary events. Sample space of this random experiment: The sample space consists of the outcome of the experiment and an outcome could be either fail in transmission or success. S={T_T), TT", IT; , TT;}

The respective probabilities are, P[112]=PTPT, = px p = p? P[T;T; ]=P[7.] P[T:] = px(1-P) =pg Since p+q=1 then q=1-p P[1,1]= P[72]P[T1] = px (1-P) = pq b) To show that the mapping from S to Sy, the range of X Let Sy is the range of X. Sx ={x} For, x = 1, 2, 3, ... Here, The first occurrence of the message in S is denoted by "x". And Sy is the mapping from S. Find the probabilities for the various values of X.

P[(TT; )(1,T;)] = P[IT:]+P[17] = pq + pq = 2 pq Let X-k>1, no message get through in time slot t-1,2,..., k-1 and a first message get through in time k. Since whether the transmission will fail or not during each slot is independent. Thus, P(No transmitted) (No transmitted),...] P(X-k,k>1) = P.:-(No transmitted) P: (transmitted) = (p)*-(p) = (p)* Therefore, the probabilities for the various values of X is p*