Continuous Random Variables

Transcription

1 Statistics 5 Lab Materials Continuous Random Variables In the previous chapter, we introduced the idea of a random variable. In this chapter we will continue the discussion of random variables. Our focus in this chapter will be continuous random variables or random variables whose values could be any of those that fall within an interval. For examples, look back at chapter. The variables we will consider will primary be things that are measured in partial units: heights, weights, lengths. We will also begin to discuss the Gaussian or Normal random variable. This random variable is the one that gives us the bell-shaped curve that is so common. Continuous Random Variables Recall the following definition of a continuous random variable. Definition a random variable is called continuous it can take any value inside an interval. The major dference between discrete and continuous random variables is in the distribution. Since the values for a continuous random variable are inside an interval, we cannot assign each value some probability. (If we did this, these probabilities would sum to infinity.) Consequently, we adopt the following solution, that area will equal probability. The way that we will describe probabilities is with areas. Suppose that H is a continuous random variable with the following distribution. f(h) The P(7<H<) is then the area under the line between 7 and, which is shaded. The line that is labeled f(h) is called the density or the probability density function and is scaled to that the total area under f(h) is. Definition: A probability density function (pdf) or density is a function that determines the distribution for a continuous random variable. The key feature of the density is that it helps us to represent probability by making probability equal to area. There are three rules that every density possesses. Three rules for probability densities for a random variable X:. The total area under the density must be.. P(a X b) 3. P(X=k) = for every k Page of 7

2 Statistics 5 Lab Materials These rules are similar to the rules that we had for distributions of discrete random variables. The first rules simply states that when you add up all the possible values that total must be. The second rules says that the probability for a range of values must be between and. Finally, the last rule means that an individual point will have probability zero. This is a result of the fact that probability is treated as area and a single point (or line) has no area. Determining f(x) is a pdf. Function f(x) is a pdf :. f (x) for all x and. f (x)dx = Determine the following function is a pdf:.5( x 4 ) f (x) = < x < ) Is f(x) for all x? Yes on the interval <x<, ( x 4 )>. Everywhere else, f(x) =. Therefore f(x) for all x.? Yes because. Is f (x)dx = f (x)dx = = dx +.5( x 4 )dx + dx = +.5(x x5 5 ) + =.5( 4 5 ) = Therefore f(x) is a pdf. Page of 7

3 Statistics 5 Lab Materials Computing Probabilities for a Continuous Random Variable The basic idea is that a probability associated with a Continuous Random Variable X is equivalent to an area under the graph of the pdf of X. For a continuous random variable with pdf f(x),. P(a < X < b) = P(a X b) = P(a < X b) = P(a X < b) = f (x)dx. P(X < b) = P(X b) = f (x)dx b 3. P(X > a) = P(X a) = f (x)dx a a b.5( x 4 ) Consider the pdf f (x) = < x < a. Find P(X < ) P(X < ) = f (x)dx = dx +.5( x 4 ) dx = +.5(x x5 5 ) =.5( 3 5 ) = 79 8 b. Find P(X>.4) Page 3 of 7

4 Statistics 5 Lab Materials P(X >.4) =.4 f (x)dx =.5( x 4 ) dx + dx.4 =.5(x x5 5 ) +.4 =.5 ( 5.4 ) ( ) =.5(.448) =.556 Theoretical Mean and Variance Suppose that random variable X is continuous with pdf f(x). The Expected Value of X (or the theoretical mean of the probability distribution of X) is E(X) = u X = x * f (x)dx The theoretical variance of the probability distribution of X is Var(X) = σ = (x µ x ) f (x)dx = x f (x)dx µ X Find the theoretical mean and variance of the random variable with pdf.5( x 4 ) f (x) = Theoretical Mean: < x < E(X) = u X = x * f (x)dx = x * dx + x *.5( x 4 ) dx + x * dx Page 4 of 7

5 Statistics 5 Lab Materials = dx +.5 (x x 5 ) dx + dx = +.5( x x 6 6 ) +. =.5( 6 6 ) =.5( 3 ) = = Theoretical Variance: Var(x) = σ = x f (x)dx µ X = x *dx + x *.5( x 4 ) dx + x * dx = dx +.5 (x x 6 ) dx + dx = +.5( x 3 3 x 7 7 ) =.5( ) =.5( 4 ) =.645 Cumulative Distribution Function The Cumulative Distribution Function (cdf) of a Continuous Random Variable species the probability that a random variable X is less than or equal to some specied value x. The CDF is usually indicated by F(x) and is defined as F(x) = P(X x) and is computed as Page 5 of 7

6 Statistics 5 Lab Materials F(x) = x f (t)dt Find the Cumulative Distribution Function (cdf) the random variable with pdf.5( x 4 ) f (x) = < x <. On the region x<, F(x) =. We begin to accumulate probability only after X exceeds.. On the interval <x< F(x) = x.5( t 4 )dt =.5(t t 5 x 5 ) =.5(x x 5 5 ) 3. After X exceeds (for x>) we have accumulated all % of the probability. So F(x) = for x>. 4. Summarizing, x F(x) =.5(x x5 ) 5 < x < x Page 6 of 7

7 Statistics 5 Lab Materials Using the cdf to compute probabilities:.5( x 4 ) Use the cdf of f (x) = < x <. Find P(X < ). Find P(X>.4) P(X <.55 ) = F(.5) =.5(.5 5 ) = P(X >.4) = F() F(.4) =.5( ) =.556 Page 7 of 7