Continuous Random Variable

Continuous Random Variable

1. Probability Density Function

   • a continuous random variable is a variable that can have any value within a given range (for example, time or height
   • the curve created by a graph showing the independent continuous random variables and the dependent probability shows the probability density function
   • if f(x) is the probability density function of the continuous random variable X, then:
     • the integral of f(x)dx between the limits of infinity and -infinity = 1
     • P(a <= X <= b) = integral f(x)dx between the limits of b and a
     • P(X = a) = 0
       • so that, for example, P(a < X < b) = P(a <= X <= b)

2. Distribution Function

   • the distribution function F(x) of a random variable X is defined by F(x) = P(X < x)
   • an expression, which gives the probability that X < a, is a function of a and is called the distribution function

3. Median, Quartiles and Percentiles

   • the median x value is the value that divides the area under the graph in half
   • the quartiles of the distribution of X are the values that divide the area under the graph into quarters
   • the 20th percentile of the distribution is the value that has 20% of the area below it (similarly for other percentiles
   • if X is a continuous random variable, then:
     • P(X < median) = 0.5
     • P(X < lower quartile) = 0.25
     • P(X < upper quartile) = 0.75
     • P(X < 20th percentile) = 0.2

4. Mean or Expected Value

   • the mean or expected value of the continuous random variable X is given by E(X) = integral of x f(x)dx
     • the integral is a definite integral, but the limits are not shown because they depend on the definition of f(x)

5. Expectation of a Function of a Continuous Random Variable

   • if g(X) is a function of X, then E(g(X)) = integral of g(x)f(x)dx
   • E(aX + b) = aE(X) + b

6. Variance and Standard Deviation

   • Var(X) = E(X x X) - [E(X) x E(X)]
     • variance = mean of the square - square of the mean
   • standard deviation = square root of the variance
   • Var(aX + b) = a x aVar(X)

7. Rectangular Distribution

   • in general, the probability density function of the rectangular distribution on the interval from a to b is given by:
     • {_1_ 0 <= x <= 5 f(x) = {b - a {0 otherwise
   • the corresponding distribution for a continuous random variable is called a rectangular distribution
   • if X has the rectangular distribution on an interval of length l, then E(X) is the mid-point of the interval and Var(X) = l x l / 12