We now introduce a new of random variable, the exponential random variable. It has a probability density function that is determined by a single parameter lambda, which is a positive number. And the form of the PDF is as shown here. Note that the PDF is equal to 0 when x is negative, which means …
Probability – L08.6 Exponential Random Variables Read More »
As an example of a mean-variance calculation, we will now consider the continuous uniform random variable which we have introduced a little earlier. This is the continuous analog of the discrete uniform, for which we have already seen formulas for the corresponding mean and variance. So let us now calculate the mean or expected value …
Probability – L08.5 Mean & Variance of the Uniform Read More »
We now start with our agenda of developing continuous counterparts of everything we have done for discrete random variables. Let us look at the concept of expectation. In the discrete case, we have defined expectation as a weighted average of the values X of the random variable, weighted according to their corresponding probabilities. In the …
Let us now give an example of a continuous random variable– the uniform random variable. It is patterned after the discrete random variable. Similar to the discrete case, we will have a range of possible values. In the discrete case, these values would be the integers between a and b. In the continuous case, any …
Probability – L08.3 Uniform & Piecewise Constant PDFs Read More »
In this segment, we introduce the concept of continuous random variables and their characterization in terms of probability density functions, or PDFs for short. Let us first go back to discrete random variables. A discrete random variable takes values in a discrete set. There is a total of one unit of probability assigned to the …
Probability – L08.2 Probability Density Functions Read More »
In this lecture, we start our discussion of continuous random variables. We will focus on the case of a single continuous random variable, and we’ll describe its distribution using a so-called probability density function, an object that will replace the PMFs from the discrete case. We will then proceed to define the expectation and the …
Probability – L08.1 Lecture Overview – Continuous RV and PDFs Read More »
By now, we have defined the notion of independence of events and also the notion of independence of random variables. The two definitions look fairly similar, but the details are not exactly the same, because the two definitions refer to different situations. For two events, we know what it means for them to be independent. …
Probability – S07.3 Independence of Random Variables Versus Independence of Events Read More »
In this segment, we will derive the formula for the variance of the geometric PMF. The argument will be very much similar to the argument that we used to drive the expected value of the geometric PMF. And it relies on the memorylessness properties of geometric random variables. So let X be a geometric random …
Probability – S07.2 The Variance of the Geometric Read More »
In this segment, we develop the inclusion-exclusion formula, which is a beautiful generalization of a formula that we have seen before. Let us look at this formula and remind ourselves what it says. If we have two sets, A1 and A2, and we’re interested in the probability of their union, how can we find it? …
Probability – S07.1 The Inclusion-Exclusion Formula Read More »
We will now study a problem which is quite difficult to approach in a direct brute force manner but becomes tractable once we break it down into simpler pieces using several of the tricks that we have learned so far. And this problem will also be a good opportunity for reviewing some of the tricks …