Page 338 - Numerical Methods for Chemical Engineering

Important probability distributions 327

Definition Cumulative probability distribution
We define from either a discrete or a continuous probability distribution the cumulative
probability distribution,
k ' x

F(X k ) = P(X j )or F(x) = p(x )dx (7.47)
j=1 x lo
that satisfies the normalization condition
F(X M ) = 1or F(x hi ) = 1 (7.48)

From the cumulative probability distribution, we can generate values of X or x at random
according to the specified probability distribution. For example, in the case of a contin-
uous variable, we generate a uniformly distributed number 0 ≤ u ≤ 1 using rand. The
corresponding random value of x, generated according to the probability distribution p(x)
satisfies
' x

F(x) = p(x )dx = u (7.49)
x lo
Definition Variance and standard deviation
We have defined the expectation for both discrete and continuous probability distributions
thatgivestheaverage,ormeanvalue,ofarandomvariable.Toobtainameasureofthebreadth
of the distribution of X, we define the variance to be the expected quadratic variation from
the mean,

2
2
var(X) = E[(X − E(X)) ] = E(X ) − [E(X)] 2 (7.50)
The square root of the variance is known as the standard deviation,

σ = var(X) (7.51)
If X and Y are independent, the variance of their sum is simply

var(X + Y) = var(X) + var(Y) (7.52)

Later we extend these definitions to multiple, interacting random variables, but first consider
some common forms of discrete and continuous probability distributions.

Bernoulli trials
Let us say that we are tossing a coin for which we have a probability p H of observing heads
and a probability p T of observing tails. Each coin toss, assumed independent, is an example
of a Bernoulli trial. If the coin is fair, p H = p T = 1/2. We now ask the following question:

If we make n independent coin tosses, what is the probability that we will observe heads n H number
of times, i.e. that our sequence of tosses will return heads n H times and tails n T = n − n HT times?

While this may seem like a question that is only important for those planning a trip to
Las Vegas, it is directly relevant to a wide variety of physical phenomena. Let us say that