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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.