Random Variables bout the random variable R- you can derive one from the other-but sometimes one is more convenient. The key point here is that neither the probability density function nor the cumulative distribution function involves the sample space of an experiment. Thus, through these functions, we can study random variables without reference to a particular experiment. For the remainder of today we'll look at three important distributions and some plication 2.1 Bernoulli distribution Indicator random variables are perhaps the most common type because of their close association with events. The probability density function of an indicator random variable b is alway PDFB(O) PDFB(1)=l-p where <p< l. The corresponding cumulative ditribution function is CDFB(O)=p CDFB(1) This is called the Bernoulli distribution. The number of heads flipped on a(possibly biased)coin has a Bernoulli distribution. 2.2 Uniform distribution A random variable that takes on each possible values with the same probability is called uniform. For example, the probability density function of a random variable U that is uniform on the set (1 PDFU(k) And the cumulative distribution function is: CDFU (k) Uniform distributions come up all the time. For example, the number rolled on a fair die is uniform on the set (1, 2, .., 68 Random Variables about the random variable R— you can derive one from the other— but sometimes one is more convenient. The key point here is that neither the probability density function nor the cumulative distribution function involves the sample space of an experiment. Thus, through these functions, we can study random variables without reference to a particular experiment. For the remainder of today, we’ll look at three important distributions and some ap￾plications. 2.1 Bernoulli Distribution Indicator random variables are perhaps the most common type because of their close association with events. The probability density function of an indicator random variable B is always PDFB(0) = p PDFB(1) = 1 − p where 0 ≤ p ≤ 1. The corresponding cumulative ditribution function is: CDFB(0) = p CDFB(1) = 1 This is called the Bernoulli distribution. The number of heads flipped on a (possibly biased) coin has a Bernoulli distribution. 2.2 Uniform Distribution A random variable that takes on each possible values with the same probability is called uniform. For example, the probability density function of a random variable U that is uniform on the set {1, 2, . . . , N} is: 1 PDFU (k) = N And the cumulative distribution function is: k CDFU (k) = N Uniform distributions come up all the time. For example, the number rolled on a fair die is uniform on the set {1, 2, . . . , 6}