Cdf of a uniform distribution
There is only one way that this can happen: both dice must roll a 1. To construct the probability distribution for X, first consider the probability that the sum of the dice equals 2. Clearly, X can also assume any value in between these two extremes thus we conclude that the possible values for X are 2,3.,12. The smallest this sum can be is 1 + 1 = 2, and the largest is 6 + 6 = 12.
the number of different values for the random variable X. However, we are interested in determining the number of possible outcomes for the sum of the values on the two dice, i.e. So we can distinguish between a roll that produces a 4 on the yellow die and a 5 on the red die with a roll that produces a 5 on the yellow die and a 4 on the red die. Notice that all 36 outcomes are distinguishable since the two dice are different colours. the value on one of the dice does not affect the value on the other die), so we see that = there are 6 ✕ 6 = 36 different outcomes for a single roll of the two dice. The two dice are rolled independently (i.e. There are 6 possible value each die can take. Recall that a function f( x) is said to be nondecreasing if f( x 1) ≤ f( x 2) whenever x 1 < x 2. The following properties are immediate consequences of our definition of a random variable and the probability associated to an event. Notice also that the CDF of a discrete random variable will remain constant on any interval of the form. Note that in the formula for CDFs of discrete random variables, we always have, where N is the number of possible outcomes of X. In other words, the cumulative distribution function for a random variable at x gives the probability that the random variable X is less than or equal to that number x. Where x n is the largest possible value of X that is less than or equal to x. Using our identity for the probability of disjoint events, if X is a discrete random variable, we can write The cumulative distribution function (CDF) of a random variable X is denoted by F( x), and is defined as F( x) = Pr( X ≤ x). Given a probability density function, we define the cumulative distribution function (CDF) as follows.Ĭumulative Distribution Function of a Discrete Random Variable