Random Variable X(1) Discrete random variable(2) Continuous random variable(3) Mixed random variableCumulative distribution function (cdf) and Survival functionMath 370X - Lecture 3Random Variables and Probability DistributionsSaumil PadhyaSeptember 19, 2016Random Variable XRandom variable is a function on a probability space S. This function assigns a real number X(s)to each sample point s∈S. A random variable is like any other function - you put a number in itand you get a number out of it.There are 3 types of random variable:(1) Discrete(2) Continuous(3) Mixed(1) Discrete random variableThe random variable X is discrete and is said to have adiscrete distributionif it can take onnumerical values only from a finite or countable infinite sequence of numbers.The probability function of a discrete random variable is denoted p(x) and is equal to the probabilitythat the value ‘x’ occurs i.e. P[X=x]. This is also sometimes referred to asprobability massfunction (pmf).0 ≤ p(x) ≤ 1Xxp(x) = 1Example 1: An ordinary single dies is tossed repeatedly and independently until the first evennumber turns up. The random variable X is defined to be the number of the toss on which thefirst even number turns up. Find the probability that X is an even number.(2) Continuous random variableThe random variable X is continuous and said to have acontinuous distributionif it can takeon numerical values from an interval of real numbers.The probability function of a continuous random variable is denoted f(x) and is referred to asprobability density function (pdf). The probability that X is in the interval (a,b) is:1P [a < X < b] =bZaf(x)dxNote that for a continuous random variable, P[X=c] = 0 for any individual point c since there canonly be probability over an interval, not at a single point. As a result, the following are all equal:P (a < X < b) = P (a < X ≤ b) = P (a ≤ X < b) = P (a ≤ X ≤ b)The pdf f(x) must satisfy the following conditions:f(x) ≥ 0∞Z−∞f(x)dx = 1Example 2:Suppose that the continuous random variable X has density function f(x) = 3 - 48x2for -.25 ≤ x ≤ .25 (and f(x) = 0 elsewhere). Find P[18≤ X ≤516].(3) Mixed random variableA mixed random variable is discrete and continuous i.e. it may have somepointswith non-zeroprobability mass combined with a continuous pdf on one or more intervals.Example 3:Suppose that X has probability of 0.5 at X = 0, and X is a continous randomvariable on the interval (0,1) with density function f(x) = x for 0 < x < 1. Find P(0 < X < 0.5),P(0 ≤ X < 0.5).Cumulative distribution function (cdf) and Survival functionGiven a random variable X, the cdf of X is:F (x) = P (X ≤ x) =xZ−∞f(x)dxThe survival function is the complement of the cdf:S(x) = 1 − F (x) = P (X > x)The probability density function can be derived from the cdf or the survival function as follows:ddxF (x) = −ddxS(x) = f (x)2For any cdf,P (a < X ≤ b) = F (b) − F (a)Example 4: The distribution function for continuous random variable U isF (u) = 1 − e−ufor u > 0Find the probability P (U ≤ 2|U >
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