I.C.1 Probability Distributions and Density. In Section
0.C (which you might want to review now), we introduced some basic
concepts of probability with the darts model. To explore these ideas
further we'll continue our examination of that model. Recall that we were
considering a circular magnetic board of radius sixty centimeters.
This magnet is very strong and uniformly attracting so that when I turn my back to it and release a magnetic dart, the dart will be drawn to the board and is equally likely to land on any point on the board.
This is our basic experiment. After performing the experiment we
measured the random variable $R$, the distance from the dart to the center
of the circle. | |

Notation: As in Chapter 0.C we will denote the probability
that the random variable $R$ has a value less than or equal to $A$ by F($A$)
where $A$ is any real number.
F is a function of $A$ called the (cumulative) probability distribution
function for the random variable $R$. When analyzed by cases we saw that $ F($A$) =
\begin{align}
0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A\le0 \\ \frac {A^2}{3600}\ \ \ \ \ \ \ \ \ 0 \lt A \lt 60 \\ 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 60 \le A \end{align}$ |

. |

**Example
I.C.1.**For our discussion, let's consider the dart falling close to
50 centimeters from the center as meaning that $R$ is in the interval [49,51],
and falling close to 10 centimeters as having $R$ in the interval [9,11].
The probabilities for $R$ being in these intervals is computed as in Section
0.C:

Point Probability Density: Recall from Section 0.C that the
average probability density (APD) of a random variable $R$ for
an interval [A,B] is the ratio of the probability that A< $R$ < B to the length of the interval $[A,B]$, i.e., |

Definition: The (point) probability density (PD) of a random
variable $R$ for a value $A$, denoted $f (A)$, measures the likelihood
that $R$ will take a value close to $A$. This number is estimated by
average probability densities for very short intervals with $A$ as one endpoint.
Let $\overline f(x)=\frac {F(x)-F(A)}{x-A} = APD(x,A)$, the APD for $R$ on an interval with endpoints $A$ and $x$ where $x$ is different from $A$. The preceding statements can be expressed by saying that as $x$ approaches $A$, the values of $\overline f(x)$ should approach $f (A)$, i.e., as $x \rightarrow A$, we should find that $\overline f(x) \rightarrow f(A)$. |

We'll
continue the example that illustrates the meaning of this by determining
the point probability density of $R$ for the values *$A*=50$ cm. and $A
= 10$ cm. [You might pause a minute here and make your own estimates for
these numbers.]

Using the probabilities from the previous discussion, we find the average probability density for $[49,50]$, $APD(49,50)= 99/3600 = 11/400$, while the $APD(50,51) = 101/3600$.

We continue to examine more closely the APD for intervals with 50 as
an endpoint in an attempt to determine whether some number can be assigned
as a point probability density to the number 50. Figure
I.17 shows a table of APD's for intervals with one endpoint being 50.

Table of APD's for intervals with one endpoint being 50

Interval | Average Prob.Density |

[49,50] | 99/3600 |

[49.9,50] | 99.9/3600 |

[49.99,50] | 99.99/3600 |

[50,51] | 101/3600 |

[50,50.1] | 100.1/3600 |

[50,50.01] | 100.01/3600 |

$= [x + 50]/ 3600$ .

Similar analysis of APD's for intervals with endpoint 10 leads to a finding that $R$ has a (point) probability density at $A= 10$ of $1/180$ or $f (10)= 1/180$. The higher probability density for $50$ in some sense explains why the value of $R$ - which measures the distance of the dart from the center of the dart board- is more likely to be closer to $50$ than to $10$.

**Comment: **It is left for the reader to find a formula for the
point probability density of $R$ for $A$ with $0 < A < 60$, *i.e.*,
a formula for f$ (A)$. [See Exercise 2.]

1. For the random variable $R$ of Example I.C.1 find the point probability density for 20. Find the point probability for 40. Explain briefly why you would expect $R$ to be close to 40 more likely than it would be close to 20.

2. For the random variable $R$ of Example I.C.1 find the point probability density for $A$ where 0 < $A$ < 1 .

**Dart Boards with Different Shapes.**

3. Suppose the dart board for our experiment is a 3' by 5' rectangle
and when the dart lands we report the distance of the dart to the
5' side with the random variable L. [See Figure I.18]

a. Find the point probability density for L at 1.

b. Find the point probability density for L at $A$ with 0 < $A$ < 3.

4. Suppose the dart board is a right triangle with legs
3' and 4' and when the dart lands we report the distance of the dart to
the 4' side with the random variable H. [See Figure I.19]

a. Find the point probability density for H at .1.

b. Find the point probability density for H at $A$ with 0 < $A$ < 3.

5. Suppose X is a random variable which takes values in the interval
[0,2] and that for each of the following definitions of F, F($A$) gives
the probability that X $A$ where 0<$A$<2.

i. Find the (point) probability density of the distribution
for X at 1.

ii. Find the (point) probability density of the distribution
for X at $A$ where 0 < $A$ < 2.

a. F(A) = .5 A. b. F(A) = 1/4 A ^{2}.
c. F(A) = A^{2} - 3/2 A.

d. F(A) = 1/8 A^{3} . e. F(A) = 1/16 A^{4}.

6. Suppose F(A) = sin(A) describes the probability that a random variable Y is less than or equal to $A$ where 0 < A < π /2. Using your calculator estimate the point probability density for Y at π/4 using the intervals [ π/6, π/4] , [π /4, π/3], and [π /4, π /4 + .1]. Discuss briefly what you think the point probability density for Y at π /4 might be. [See problem 11 in section 0.C.]

7. Suppose that an object is inside a sphere of radius 10 centimeters
and it is equally likely that it is at any point in the sphere. Let $R$ denote
the random variable that measures the distance from the object to the center
of the sphere. [See problem 12 in section 0.C.]

Why is the probability that $R$ <= A with A < 10 given
by A^{3 }/1000? Find a formula for the point probability density
for $R$ at $A$ with 0 <$A$ <10.

8. In Example I.C.1 consider the average probability density for intervals of the form [x,60] and [60,x]. Discuss the issue of whether there is a point probability density for $R$ at $A$=60 based on some calculations for these intervals.

9. **Another Darts Board**. Suppose the dart board has a shape bounded
by the polygon in Figure *** in the coordinate plane and when the dart
lands we report the first coordinate of the point measuring the distance
of the point to the Y axis with the random variable X.

a. Draw a mapping diagram and sketch of what the
cumulative distribution function F for X might be, based on the figure,
i.e., the function F where F(A) is the probability that $X \le A$.

b. Estimate the point probability density for X at $A$ = 1, 2,
3, and 4..

c. Find the point probability density for X at $A$ with 0 < $A$ < 5.

10. **Point Probability Density and The Tangent Problem**. Write
a comparison of the treatment of point probability density to the
solution of the tangent problem. Discuss the following statement in your
essay: " The probability density of a random variable X at a point $A$ is
measured by the slope of the line tangent to the graph of the cumulative
distribution function F at the point $(A,F(A))$."

11. Project on Estimation and Archimedes: The Greek mathematician and
scientist Archimedes (287-212 B.C.E.) wrote perhaps the first work to give
a systematic method for estimation. Read *On the Measurement of the Circle*
where Archimedes estimates pi in Proposition 3. Write a paper describing
the content and method of this work.