If we flip two fair coins, we can with equal probability obtain head on the first and heads on the second (H H), heads on the first and tails on the second (H T), tails on the first in heads on the second (T H), or tails on the first and tails on the second (T T).

• We therefore see that the probability of obtaining 0 Heads on two flips is 1/4, the probability of obtaining 1 Heads is 2/4 = 1/2, and the probability obtaining two Heads is 1/4.
• We can represent these probabilities with the histogram indicated below.
• The total area under the histogram is 1, which is of course the sum of all the probabilities.
• When the total area under a graph representing relative probabilities is 1, we say that the graph is normalized.

The graph is set up with bars of length 1.

• The first bar, representing the probability of 0 Heads, has x = 0 in the middle and therefore extends from x = -1/2 to x = 1/2.
• The second bar represents the probability of 1 Heads and is therefore centered at x = 1, extending from x = 1/2 to x = 3/2.
• The third bar represents the probability of 2 Heads and is therefore centered at x = 2, extending from x = 3/2 to x = 5/2.

The probabilities for three coins are as indicated below.

• Note that the eight equal probabilities when we flip three coins are the probabilities of HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.
• The histogram below represents the probabilities of obtaining 0, 1, 2 or 3 Heads.

If we flip greater in greater numbers of coins, we obtain histogram like the one below.

• The most likely number of heads for a flip of n coins is n/2.
• When the number n is large, the likelihood of obtaining numbers near 0 or near n is very small and histogram effectively disappears into the x axis.

If we adjust the vertical scale of our graph so that the total area under the graph is 1, then the graph will be normalized.

• If we were to shift the graph to the left n/2 units, the peak of the graph would be at x = 0, and x would represent the number of heads above or below the most likely value x = n/2.

Video Clip #01

As the number n of coins gets very large, the normalized distribution histogram representing the number of Heads above or below the most likely number n/2 approaches the normal distribution function p(x) = 1 / `sqrt(2 `pi) * e ^ (- x^2 / (2 `sigma^2) ).

• This function is normalized in that its integral, from - infinity to infinity, is 1.

For a normal distribution function, the probability of that occurs between x = a and x = b is represented by the area under the curve between a and b.

• This probability is represented by the indicated integral.
• Note that p(x) itself does not denote a probability; probabilities are indicated by areas, and p(x) is just the height of the graph.
• To get a probability we must therefore find and area by integrating p(x) over the appropriate interval.

Returning to the first example, where we flip two coins, we can construct a graph of the cumulative probability function.

• The cumulative probability is the probability that the number of heads will be less than or equal to a certain number.

• In our example, the probability of getting <= 0 Heads is just the probability of getting 0 Heads, or 1/4.
• The probability of getting <= 1 Heads is the probability of getting either 0 or 1 Heads, which is 1/4 + 1/2 = 3/4.
• The probability of getting <= 2 Heads is the probability of getting 0, 1 or 2 Heads, which is 1.

• These probabilities are indicated in the second graph below.

For a typical normalized probability distribution function p(x), like the normal distribution function indicated below, the probability of getting <= x is the area to the left of x, denoted p(-infinity < t < x ).

• This probability is just the integral from -infinity to x of p(t).
• We denote the resulting cumulative probability distribution function by P(x).
• P(x) is defined by the integral in the figure below.
• We note that since the integral of p(t) from -infinity to infinity is 1, P(x) will approach the total integral, which is 1.

• Our graph will therefore be asymptotic to y = 1, as indicated.

Video Clip #02

The probability of an occurrence between 0 and T, for some probability distribution function defined for x >= 0, is indicated by the shaded area on the graph of p(x) below.

• Tmedian is defined as the value of T for which the probability p(0 < x < T) is 1/2.

Video Clip #03

If we wish to find the average age of a group of people, we add the ages of the individuals then divide by the number of individuals.

• Since we have used to probability distribution function instead of actual numbers of people, the total number of people will be the integral of the probability distribution function, which is just 1.

• The mean age is obtained by dividing the sum of all the ages (the first-moment integral) by the number of people (the sum 1 of all the probabilities), obtaining the result in the second-to-last line below.
• Note the similarity between this expression and the expression for the center of mass.

The final result is that, when the probability distribution function is normalized, the average age is simply the first-moment integral indicated in the last line below.

The cumulative distribution function is obtained from the integral below, and is found to be P(x) = 1 - e^(-.14 x).

If p(x) represents the probability distribution of the lengths of time required for a certain task, then we can find the mean length of time by calculating the first-moment integral indicated in the figure below.