Previous Sleep Stage Current Stage Non-REM REM Non-REM REM Wake 160 1,733 175 358 6,079 31,880 7,609 346 1,588 33,814 Totals 8,127 987 Based on Caffo, B. S, et al. "An overview of observational sleep research with application to sleep stage transitioning." Chance, Vol. 22, No. 1, Winter 2009 (Table 2). c. Are the events (previous stage 1s stage is REMI mutually exclusive? stage is REM independent? REM) and (curned , Leurred d. Are the events (previous stage is REM) an e. Are the events

Refer to the sleep apnea study published below. Recall that the various stages of sleep for a large group of sleep apnea patients were monitored in 30-second "epochs", and the sleep stage for the previous and current epoch was determined.
a). Given that the previous sleep stage for the epoch was the Wake state, what is the probability that the current sleep stage is REM?
b). Given that the current sleep for the epoch is REM, what is the probability that the previous sleep stage was not the wake state?
c). Are the events (previous stage is REM) and (current stage is REM) mutually exclusive?
d) Are the events (previous stage is REM) and (current stage is REM) independent?
3). Are the events (previous stage is Wake) and (current stage is Wake) independent?