Physics 3700 Durkin / Leonard Spring, 2016

Problem Set 1 (Due Monday 02/01/16)

1. Data Set Mean & Standard Deviation

Taylor, Problem 4.2, page 111. Calculate directly, and then check using a calculator (perhaps online) having statistics functions.

2. Effect of Transformations on Distributions

Find the results and show proofs for the following transformations. • Add a constant c to each element 𝑥! in a sample (𝑖 = 1,𝑛) giving resulting elements 𝑦! = 𝑥! + 𝑐.

Relate the resulting mean and variance (for y) to the original mean and variance (for x). • Multiply each element 𝑥! by the constant c to give elements 𝑦! = 𝑐 ∙ 𝑥!. Again relate the resulting

mean and variance to the original quantities.

3. Dice Sum Distribution

The probability distribution describing the sum of the dots (𝑥) showing on a pair of six-sided dice is:

𝑃 𝑥 =𝑥 − 136        for    𝑥 = 2, 3, 4, 5, 6, 7            and                  𝑃 𝑥 =

13− 𝑥36      for    𝑥 = 8, 9, 10, 11, 12

• Find the mean and variance (and thus the standard deviation) of this distribution. Calculate directly according to the definitions of mean and variance for a discrete distribution.

• Now calculate the variance by using the following theorem. The variance of the sum of independent random variables is equal to the sum of their variances.

Note that this probability distribution is used in Lab 1.

4. Neutrino Statistics

A detector located underground in a salt mine near Cleveland detected a burst of 8 neutrinos at the same time as the optical observation of Supernova 1987A. Use Poisson statistics to answer the following questions. Also, offer a brief interpretation of the results. • Find the probability of observing 8 or more neutrinos in one day, if, on average, the detector

would normally find 2 neutrino interactions per day. • Find the probability of observing 8 or more neutrino interactions in a ten-minute time interval.

Again, assume that the experimenters expected, on average, 2 neutrino interactions per 24 hours. (This is what was observed!)

Note that in this latter calculation, the probability of observing 9 or more events is much less than that for 8 events. So, to a very good approximation, the probability of 8 or more is simply that for 8.

Physics 3700 Durkin / Leonard Spring, 2016

PS 1

5. Particle Decay

The probability density function describing the time (𝑡) between the creation and decay of a certain unstable elementary particle is given by:

𝑓 𝑡 = 0        for      𝑡 < 0            and            𝑓 𝑡 = 𝑎𝑒!! !        for        𝑡 ≥ 0        with            𝜏,𝑎  constant

• Using normalization condition (Eqn. 5.13, p. 128), find the normalization constant (𝑎) in terms of the lifetime (𝜏). What are the appropriate limits of the integration?

• Find the average time it takes for a particle to decay. • Find the variance of this PDF. • What is the probability for a particle to live greater than three lifetimes?

6. Probabilities in Dice Throwing

Taylor, Problem 10.3, page 241. Note that wording in the textbook is somewhat ambiguous. Read as either 1 throw of 4 dice or, the equivalent, 4 throws of 1 die, but not 4 throws of 4 dice.

7. Survival Probabilities

Taylor, Problem 10.10, page 242.

8. Developing Probability Theory

In the 1650’s Chevalier de Méré and Blaise Pascal (with Pierre de Fermat acting as a consultant) discussed the following question (while developing the foundations of probability theory). Which of the following is more likely?

A. In one throw of four dice to get at least one six. B. In twenty-four throws of two dice to get at least one double-six.

Explain why you think A or B is correct. Since 4/6 = 24/36, then the naïve answer might be that they are equal. Explain why this reasoning is flawed. (Although Pascal and Fermat, in particular, were smart people, this and similar questions were tricky to figure out at this early time.)