Engineering Texts

We will now talk about conditional expectations of one random variable given another. As we will see, there will be nothing new here, except for older results but given in new notation. Any PMF has an associated expectation. And so conditional PMFs also have associated expectations, which we call conditional expectations. We have already seen …

Probability – L07.3 Conditional Expectation & the Total Expectation Theorem Read More »

We have already introduced the concept of the conditional PMF of a random variable, X, given an event A. We will now consider the case where we condition on the value of another random variable Y. That is, we let A be the event that some other random variable, Y, takes on a specific value, …

Probability – L07.2 Conditional PMFs Read More »

In this last lecture of this unit, we continue with some of our earlier themes, and then introduce one new notion, the notion of independence of random variables. We will start by elaborating a bit more on the subject of conditional probability mass functions. We have already discussed the case where we condition a random …

Probability – L07.1 Lecture Overview – Conditioning of Random Variable; Independence of r.v.’s Read More »

Let us now revisit the subject of expectations and develop an important linearity property for the case where we’re dealing with multiple random variables. We already have a linearity property. If we have a linear function of a single random variable, then expectations behave in a linear fashion. But now, if we have multiple random …

Probability – L06.8 Linearity of Expectations & The Mean of the Binomial Read More »

By this point, we have discussed pretty much everything that is to be said about individual discrete random variables. Now let us move to the case where we’re dealing with multiple discrete random variables simultaneously, and talk about their distribution. As we will see, their distribution is characterized by a so-called joint PMF. So suppose …

Probability – L06.7 Joint PMFs and the Expected Value Rule Read More »

We will now work with a geometric random variable and put to use our understanding of conditional PMFs and conditional expectations. Remember that a geometric random variable corresponds to the number of independent coin tosses until the first head occurs. And here p is a parameter that describes the coin. It is the probability of …

Probability – L06.6 Geometric PMF Memorylessness & Expectation Read More »

An important reason why conditional probabilities are very useful is that they allow us to divide and conquer. They allow us to split complicated probability modes into simpler submodels that we can then analyze one at a time. Let me remind you of the Total Probability Theorem that has his particular flavor. Total Probability Theorem …

Probability – L06.5  Total Expectation Theorem Read More »

We now move to a new topic– conditioning. Every probabilistic concept or probabilistic fact has a conditional counterpart. As we have seen before, we can start with a probabilistic model and some initial probabilities. But then if we are told that the certain event has occurred, we can revise our model and consider conditional probabilities …

Probability – L06.4 Conditional PMFs & Expectations Given an Event Read More »

In this segment, we will go through the calculation of the variances of some familiar random variables, starting with the simplest one that we know, which is the Bernoulli random variable. So let X take values 0 or 1, and it takes a value of 1 with probability p. We have already calculated the expected …

Probability – L06.3 The Variance of the Bernoulli & The Uniform Read More »

We have introduced the concept of expected value or mean, which tells us the average value of a random variable. We will now introduce another quantity, the variance, which quantifies the spread of the distribution of a random variable. So consider a random variable with a given PMF, for example like the PMF shown in …

Probability – L06.2 Variance Read More »