   Next: Wave-Particle Duality Up: Continuous Probability Distributions Previous: Continuous Probability Distributions

### Exercises

1. In the game'' of Russian roulette, the player inserts a single cartridge into the drum of a revolver, leaving the other five chambers of the drum empty. The player then spins the drum, aims at his/her head, and pulls the trigger.
1. What is the probability of the player still being alive after playing the game times?
2. What is the probability of the player surviving turns in this game, and then being shot the th time he/she pulls the trigger?
3. What is the mean number of times the player gets to pull the trigger?

2. Suppose that the probability density for the speed of a car on a road is given by where . Here, and are positive constants. More explicitly, gives the probability that a car has a speed between and .
1. Determine in terms of .
2. What is the mean value of the speed?
3. What is the most probable'' speed: i.e., the speed for which the probability density has a maximum?
4. What is the probability that a car has a speed more than three times as large as the mean value?

3. An radioactive atom has a uniform decay probability per unit time : i.e., the probability of decay in a time interval is . Let be the probability of the atom not having decayed at time , given that it was created at time . Demonstrate that What is the mean lifetime of the atom?   Next: Wave-Particle Duality Up: Continuous Probability Distributions Previous: Continuous Probability Distributions
Richard Fitzpatrick 2010-07-20