Download Discrete and Continuous Random Variables

Discrete and Continuous Random Variables
Discrete : A random variable is called a discrete random variable
if its set of possible outcomes is countable. This usually occurs for
any random variable which is a count of occurrences or of items, for
example, the number of large diameter piles selected in the previous
example.
Continuous : A random variable is called a continuous random
variable if it can take on values on a continuous scale. This is
usually the case with measured data, such as cohesion.
Examples:
1) Let X be the number of blows in an Standard Penetration Test
-- X is discrete.
2) Let Y be the number of piles driven in one day
-- Y is discrete.
3) Let Z be the time till consolidation settlement exceeds some threshold
-- Z is continuous.
4) Let W be the number of grains of sand involved in a sand cone test
-- W is discrete, but is often approximated as continuous,
particularly since W can be very large.
Example:
Suppose that the median structural capacity of a pile, under certain lateral support
conditions, is 60 kN. Three piles are selected randomly for testing. Let X be the
number of piles having strength under 60 kN, from amongst the three selected.
The random variable X can have value 0, 1, 2, or 3. If S is the event that a pile has
strength exceeding 60 kN, and F is the event that its strength is less than 60 kN,
then X is the number of F’s and P[ F ] = P[ S ] = ½, so that
Sample Space
Probability
X
Probability Distribution
SSS
SSF
SFS
FSS
SFF
FSF
FFS
FFF
1/8
1/8
1/8
1/8
1/8
1/8
1/8
1/8
0
1
1
1
2
2
2
3
P[ X = 0 ] = 1/8
P[ X = 1 ] = 3/8
P[ X = 2 ] = 3/8
P[ X = 3 ] = 1/8
Continuous Random Variables
Consider the probability that a grain silo experiences a bearing capacity failure exactly
T = 4.3673458212 … after it was installed.
Clearly this probability is vanishingly small!
e.g.,
P[ T = 4.3673458212…] → 0
We could however write the probability of the lifetime over a time range as
P[ t < T < t + dt ] = fT (t ) dt
where fT (t ) is called the probability density function of T . It gives the relative
likelihood that T = t (or, more specifically, that t < T < t + dt ).
The word "density" is used because "density" must be multiplied by a length in order
to get a "mass".
The function f (t ) is now the relative likelyhood that T lies in a very small interval near t.
Roughly speaking we can now think of this as P[T = t ] = f (t ) dt
Probabilities are now obtained as areas under f (t ).
Definition :
The function f ( x) is a probability density function for the
continuous random variable X , defined over the set of real numbers, if
1) 0 ≤ f ( x) < ∞, for all -∞ < x < ∞,
∞
2)
∫
f ( x) dx = 1 (the area under the PDF is unity)
−∞
b
3) P[a < X < b] =
∫ f ( x) dx
a
NOTE : it is important to recognize that, in the continuous case,
f ( x) is not a probability. It has units of probability per unit length.
In order to get probabilities, we have to find areas under the pdf,
ie. sum up values of f ( x) dx.