The conditional probability distribution of Y given X=x is fY |x(y)=x e^-x y for y>0, and the ma

The conditional probability distribution of Y given X=x is...

Key Concepts

Joint Distribution

The joint distribution of two or more random variables represents the probability that each of the variables falls within a particular range simultaneously. It can be constructed using the conditional distribution and the marginal distribution, encapsulating the complete probabilistic information about the relationship between the variables.

Uniform Distribution

A uniform distribution is a probability distribution where all outcomes are equally likely within a given interval. In the context of continuous variables, this implies that the probability density is constant within the interval. This concept provides a simple model for uncertainty when there is no additional information favoring any outcome within a specified range.

Expectation

Expectation, or the expected value, is a fundamental concept in probability that provides a measure of the central tendency or mean of a random variable. In conditional contexts, the conditional expectation gives the mean of the random variable when the value of another variable is known, offering a refined understanding of its distribution under certain conditions.

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time between events in a Poisson process. It is characterized by a rate parameter, and its density function can be expressed in terms of this parameter. When a conditional distribution of one variable given another is exponential, it means that, for each fixed value of the conditioning variable, the distribution of the other variable follows an exponential pattern.

Conditional Probability Distribution

A conditional probability distribution specifies the probability of an event or outcome given that another event has occurred. It is used to model the behavior of one random variable when information about another variable is known. This concept is central in probability theory and statistical inference, particularly when exploring dependencies between variables.

Marginal Distribution

The marginal distribution of a random variable is obtained by integrating or summing the joint probability distribution over the other variables. This process, called marginalization, extracts the distribution of a subset of variables without reference to the others, providing insight into the behavior of that individual variable within a joint distribution.

Transcript

00:01 In this question, we have two random variable x and y, and we are given the conditional density function for y at x.

00:22 Let's say it is equal to x times e to the power negative xy, y is greater than zero and it is equal to zero else.

00:38 Okay, under this condition, we want to find the probability of y less than two, conditioning on x being two.

00:54 Okay, then by our definition, it is actually equal to the integral or y less than, the region determined at y less than two, and the integral over the conditional function for y conditioning on x being two, dy.

01:19 Okay, so the first thing for us is to plug two into our expression to find this density.

01:31 And by our previous expression, we know this density is equal to, now as x is equal to two times e to the power, by negative two y.

01:46 Okay, and by our definition, y need to be greater than zero, so the integral goes from zero to two, two times e to the power negative two y, dy, which is equal to e to the power negative two y, and y evaluated at two and zero, one minus e to the power negative four.

02:17 Okay, this is the first part.

02:21 The second part, we are required to compute the conditional expectation for y under this guy.

02:29 The same thing, this is the integral for the whole space.

02:35 I mean, y goes from zero to positive infinity.

02:37 Y times this density, okay, which is equal to two times e to the power negative two y, dy.

02:52 And now combine those long term, we know it is equal to d negative e to the power minus two y.

03:04 And use the integration by part, times minus y evaluated at positive infinity and zero, minus negative e to the power dy.

03:27 Okay, when y goes to positive infinity, this term goes away, because this term is much smaller than this guy.

03:36 So the whole term goes away.

03:38 Y goes to zero, then this term goes away.

03:42 So we get, doing some conservation, zero to positive infinity, e to the power negative two y, dy.

03:54 Add a two here and divide it back.

04:00 Now this term is equal to times negative e to the power negative two y, y evaluated at positive infinity and zero.

04:26 So it is equal to one over two.

04:27 The whole thing is equal to one over two.

04:37 Let me just do something to give us more space.

04:55 For the fourth part, we are required to compute the conditional expectation for any number x.

05:13 Okay, by our assumption, as x is a uniform distribution on zero or ten.

05:22 So we know here, we need to assume x to be greater than zero and less than ten.

05:31 Okay, and this expectation is only meaningful in this range.

05:36 Okay, by the definition, this is equal to, integral goes from zero to positive infinity, y times the conditional density, right? okay, that is equal to y times x, times e to the power negative xy, dy.

06:04 Okay, do the same thing as we just want to integrate everything with respect to y.

06:09 So we can regard x as a constant.

06:12 Combining these terms again, it is equal to d negative e to the power negative xy.

06:21 Okay, then using integration by power, y times negative e to the power negative xy.

06:30 Well, y is positive infinity and zero and plus, because we can do some consolation with the minus sign, just as what we've done previously.

06:41 So it can be written as e to the power negative xy, dy.

06:50 Okay, notice x is greater than zero by our assumption.

06:55 So when y goes to positive infinity, this term just goes away.

07:00 When y goes to zero, then this term is equal to zero...

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