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Math 3680 Lecture #8 Continuous Random Variables Suppose we throw a dart at a number line in such a way that it always lands in the interval [1,3]. Let X be the number that the dart hits. Since the possible values of X consist of an entire interval, we call X a continuous random variable. P( c X d ) Discrete Continuous Definition: A continuous random variable X on an interval [a, b] has a probability density function (pdf) f ( x) that satisfies the following three conditions: 1. f ( x ) ≥ 0 for all values of x. b 2. f ( x)dx 1. a d 3. P (c X d ) f ( x)dx. c The cumulative distribution function (cdf) of X is FX ( x ) = P( X x ). Example: Let X be a random variable with density function f ( x ) = (3 / 14) (x + x2) over [0,2]. A) Verify that f ( x ) is a density function. B) Compute P ( X 1). C) Compute F( x ) for x < 0, for 0 x 2, and for x > 2. Graph of cumulative distribution function 1 0.8 0.6 0.4 0.2 0.5 1 1.5 2 2.5 3 Properties of cumulative distribution functions • F( x ) = P( X x ) • 0 F( x ) 1 • F is non-decreasing • F is right-continuous: F(x+) = F(x) for each x • For a continuous r.v. X, F is always continuous • For any a < b, P( a < X b ) = F( b ) - F( a ) b f ( x)dx a Theorem: F ’ ( x ) = f ( x ). x Proof. By definition, F ( x) f (t )dt. Therefore, by the Fundamental Theorem of Calculus, F ’ ( x ) = f ( x ). Definition: For a random variable X, the r th percentile (denoted by x r/100) is the value x so that r F ( x) . 100 The idea is that r % of the area under the curve lies to the left of x r/100. Example: Suppose a random variable X has pdf 1 f(x)= 2 x over (1, ). Find the median (50th percentile) of the distribution. Definition: In statistics, we will often have occasion to compute the critical value that corresponds to a predetermined significance level. The significance level, denoted by a, sets a desired probability, or an area under a tail of the pdf. This kind of calculation is entirely equivalent to finding percentiles. Example: Suppose a random variable X has pdf 1 f(x)= 2 x over • Find the right-tail critical value if the significance level is a = 0.01. Moments Definition: m E ( X ) x f ( x)dx E( X ) 2 x f ( x ) dx 2 Var( X ) = E[ (X - m)2 ] s = SD( X ) = Var( X ) Var( X ) = E[ X 2 ] - m2 Theorem: If a and b are real constants, then E( a X + b ) = a E( X ) + b Var( a X + b ) = a2 Var( X ) SD( a X + b ) = | a | SD( X ) Proof: Example: Suppose a random variable X has pdf f ( x ) = 6 x - 6 x2 over [0,1]. Find the mean, variance, and standard deviation of X. 1.4 1.2 1 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1 Example: A random variable X is said to have the c2(4) distribution if its pdf is x x / 2 f ( x) e , x 0. 4 0.175 0.15 0.125 0.1 0.075 0.05 0.025 2 4 6 8 Find the mean and standard deviation of X. 10 12 14 The Uniform Distribution UNIFORM DISTRIBUTION A continuous random variable X is said to have a Uniform(a, b) distribution if its density function is given by 1 f ( x) ba for a x b. Example: Compute the mean and standard deviation of the Uniform(a, b) distribution. The Exponential Distribution EXPONENTIAL DISTRIBUTION A continuous random variable X is said to have an Exponential(q ) distribution if its density function is given by f ( x) 1 q for x 0. e x /q Exercise: Confirm that f ( x) 1 q e x /q is a probability density function (for x 0). Exercise: Compute the mean, variance, and standard deviation of the Exponential(q ) distribution. Memoryless Property of the Exponential Distribution Theorem: Let X be an exponential random variable. Then for all t ≥ 0 and all s ≥ 0, P( X t s | X t ) P( X s) Proof: Example: In the Luria-Delbrück mutation model, it is assumed that a certain population experiences 0.25 mutations per hour. A mutation just occurred. Compute the probability that a) At least four hours pass until the next mutation occurs. b) At least four hours, but not more than 8 hours pass until the next mutation occurs. c) Repeat (a) and (b), but given that the last mutation occurred three hours ago.