• Home
  • Textbooks
  • A Student's Guide to Data and Error Analysis
  • Probability distributions

A Student's Guide to Data and Error Analysis

Herman J. C. Berendsen

Chapter 4

Probability distributions - all with Video Answers

Educators

Chapter Questions

01:14

Problem 1

In a lottery 5 percent of the tickets will produce a prize. If you buy ten tickets, what is the probability that you obtain no prize, 1 prize, 2 prizes, ...? Assume that there are so many tickets and prizes that the probability of obtaining a prize does not depend on the number of prizes you already have (this is called: a lottery with replacement).

Adam Harper
Adam Harper
Numerade Educator
01:25

Problem 2

When it is known that one measurement $x$ has a probability of exceeding a given value $x_m$ of 1 percent what then is the probability that at least one measurement in a series of 20 independent measurements will exceed $x_m$ ?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
View

Problem 3

There will be elections where voters can elect one of two presidential candidates. You want to perform an opinion poll and predict the outcome with a standard uncertainty of 1 percent. You expect roughly equal votes for either candidate. Assume that you are able to obtain the opinion of an unbiased random selection of voters, how many people do you have to select (what should be your sample size)?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
View

Problem 4

You observe $n$ independent events, each of which can have an outcome of 0 or 1. You count $k_0$ zeros and $k_1$ ones $\left(k_0+k_1=n\right)$.
(a) What is your best estimate of the probability that a one appears?
(b) Give an estimate for the standard uncertainty in $k_0$.
(c) What is the standard uncertainty in $k_1$ ?
(d) You are finally interested in the ratio $r=k_1 / k_0$. What is the standard uncertainty in $r$ ?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
View

Problem 5

Show that the Poisson function 4.33 is normalized.

Rashmi Sinha
Rashmi Sinha
Numerade Educator

Problem 6

(a) With the hospital example of Fig. 4.4: assume each patient occupies a bed for one day and the ward has seven beds. When more than seven patients arrive, the excess is transported to another hospital. How many beds are occupied on average?
(b) How many patients per day are transported on average?
(c) If an unoccupied bed costs $$\$ 300$$ per day and transporting one patient costs $$\$ 1500$$, financially optimize the number of beds. How many patients per day are transported in the optimized case?

Check back soon!

Problem 7

(a) A photosensitive device produces one electrical impulse for every absorbed photon, but also produces impulses when there is no light (the "dark current"). The number of impulses counted in 1 s is 100 without radiation and 900 with radiation. How large is the relative standard uncertainty in the measured radiation intensity?
(b) How large will be the relative standard uncertainty in the measured radiation intensity when the measurement (with and without radiation) is repeated 100 times?

Check back soon!
02:30

Problem 8

What is the probability that a sample from a normally distributed quantity lies in the interval $[\mu-0.1 \sigma, \mu+0.1 \sigma]$ ?

Robin Corrigan
Robin Corrigan
Numerade Educator
01:31

Problem 9

(See the data sheet NORMAL DISTRIBUTION on page 205)
Using the approximation for large $x$ mentioned on page 2 of the data sheet NORMAL DISTRIBUTION, determine the probability that the value $x=6 \sigma$ is exceeded. Is this approximation valid for this case?

Tyler Moulton
Tyler Moulton
Numerade Educator

Problem 10

The central limit theorem has a useful application: By adding 12 random numbers $r$, which are uniformly distributed over the interval $[0,1\rangle$, and subtracting 6 from the sum, you obtain in a good approximation a sample from a normal distribution with $\mu=0$ and $\sigma=1$ :
$$x=\sum_{i=1}^{12} r_i-6$$
(a) Show that $\left\langle x^2\right\rangle=1$.
(b) Generate a list of 100 normally distributed numbers by this method.
(c) Plot the cdf of this list on a "probability" scale.

Check back soon!
04:18

Problem 11

Compute the mean and variance of the exponential distribution.

Robin Corrigan
Robin Corrigan
Numerade Educator

Problem 12

Refer to the example on page 49. In a similar trial the following results were obtained:
treatment group: $-6,2,-8,-7,-12$
control group: $5,-1,3,-4,0$
Compute the F-ratio and the corresponding cumulative probability using the F-distribution. What conclusions would you derive from this F-test?

Check back soon!