- Probability and Uncertainty
- Probability in Physics
- Discrete Random Variables
- Random Walk
- Continuous Random Variables

We all know that if we have a fair coin, it is equally likely that H or T may result from each throw. Hence, for a fair coin we have that . Hence, for a fair coin .

Recall we are interested in obtaining heads in throws regardless of what sequence they appear in. For example, all the heads can occur, for example, in the last throws, as well as in any sequence of heads in throws. Hence, we need to find out how

Hence the probability of obtaining -heads in -throws is given by

A random variable having as its possible outcomes, with the probability for the outcomes given by eq.(4.6) above is called a

(4.9) | |||

(4.10) |

The result above is simply a statement that when we throw a coin times, we are certain that the outcome will either be 0 (no) head, or 1 head, or 2 heads all the way to all heads, that is, heads. For our example of tossing a coin three times we have for the probability .

The crucial point to note above is that the proportionality constant is

For a particle undergoing a random walk, its position at every
point in its
-steps is a random variable. An important tool for studying the
behaviour
of random variables is to compute the **average values** of
quantities of interest. For a function of the random variable , say
, let us denote its average value by . We then have

(4.13) |

(4.14) |

The two most important properties of any random variable is its average and its standard deviation. Let us return to the random walk with . The average position of the particle after -steps is given by

(4.15) | |||

(4.16) |

The reason we get zero is because we have assumed equal probability to step to the right or to the left. Hence, on the average, its steps on either side of the origin cancel, with the average being at the starting point. However, we intuitively know that even though the average position of the particle undergoing random walk is zero, it will deviate more and more from the origin as it takes more and more steps. The reason being that every step the particle takes is random, it is highly unlikely that the particle will take two consecutive steps in opposite directions. The measure of the importance of the paths that are far from the origin is measured by the average value of the

We have the important result from (4.21) and (4.24)that, since, for , , we have the following

The equation above has an important interpretation. In any particular experiment, all we can obtain is = number of heads for trial. So how do we compute ? We would like to set , but there are errors inherent in this estimate, since in any particular set of throws, we can get any value of which need not be equal to . In other words, what is the error we make if set ? Eq. (4.26) tells us that for a fair coin, with , if we compute , we have

(4.23) | |||

(4.24) |

In other words, the estimate that we obtain for from our experiment, namely is, to within errors which are approximately , equal to the actual value. The point to note that the errors that are inherent in any estimate are quantified above, and go down as the ,where sample-size. In general, for any random variable with standard deviation given by , the estimate for the probability , where is the number of times that the outcome has occurred, is given by

In general, let be an estimate of some quantity that has a standard deviation given by , and derived from a sample of size . The generalization of eq.(4.29) states the following.

(4.26) | |||

(4.27) |

The relation of with what it is estimating, namely , is graphically shown in Figure4.3.

(4.28) | |||

(4.29) |

(4.30) |

(4.31) | |||

(4.32) | |||

(4.33) |

The humble uniform random variable , it turns out surprisingly, is one of the most important random variables. The reason being that one can prove a theorem that

(4.34) | |||

(4.35) | |||

(4.36) | |||

(4.37) | |||

(4.38) |

As mentioned earlier, a model for diffusion is to consider a particle doing a random walk in a (continuous) medium. It can be shown that its probability distribution is then given by the normal distribution. Suppose the particle starts its random walk at time from the point ; then at time its position can be anywhere in space, that is . The probability for it to at different values of is given by

(4.39) |