A small-business Web site contains 100 pages and 60 %, 30 %,...

A small-business Web site contains 100 pages and $60 \%$, $30 \%,$ and $10 \%$ of the pages contain low, moderate, and high graphic content, respectively. A sample of four pages is selected without replacement, and $X$ and $Y$ denote the number of pages with moderate and high graphics output in the sample. Determine: (a) $f_{X Y}(x, y)$ (b) $f_{X}(x)$ (c) $E(X)$ (d) $f_{Y \mid 3}(y)$ (e) $E(Y \mid X=3)$ (f) $V(Y \mid X=3)$ (g) Are $X$ and $Y$ independent?

A small-business Web site contains 100 pages and $60 \%$, $30 \%,$ and $10 \%$ of the pages contain low, moderate, and high graphic content, respectively. A sample of four pages is selected without replacement, and $X$ and $Y$ denote the number of pages with moderate and high graphics output in the sample. Determine:
(a) $f_{X Y}(x, y)$
(b) $f_{X}(x)$
(c) $E(X)$
(d) $f_{Y \mid 3}(y)$
(e) $E(Y \mid X=3)$
(f) $V(Y \mid X=3)$
(g) Are $X$ and $Y$ independent?

Key Concepts

Independence of Random Variables

Two random variables are independent if the joint probability mass function can be expressed as the product of their marginal probability mass functions. In sampling without replacement, checking for independence involves verifying whether the outcome of one variable affects the probability distribution of the other.

Conditional Distribution

This concept involves the probability distribution of one random variable given that another random variable has a specific value. It helps in understanding the dependency structure between variables and is computed by adjusting the joint probabilities by the marginal probability of the given condition.

Expectation and Variance of a Random Variable

Expectation (or mean) is the long-run average value of a random variable computed as a weighted sum of its possible values, while variance measures the spread or dispersion around this mean. These concepts are key in summarizing the central tendency and variability in the outcomes of random experiments, including those involving hypergeometric distributions.

Marginal Probability Mass Function

The marginal probability mass function is obtained by summing the joint probability mass function over the values of one or more variables. It reflects the probability distribution of a subset of the random variables independent of the others and is crucial for understanding the behavior of one variable in isolation within a joint distribution.

Joint Probability Mass Function

This function gives the probability that two or more discrete random variables simultaneously take on specific values. In problems concerning sampling from a finite set without replacement, the joint mass function helps to determine the likelihood of observing a particular combination of outcomes across the categories.

Multivariate Hypergeometric Distribution

An extension of the hypergeometric distribution, the multivariate version handles situations where the population is classified into more than two categories. It calculates the joint probability for counts of items in several categories simultaneously and is useful when dealing with multiple classifications in a single sampling experiment.

Hypergeometric Distribution

This concept describes the probability of drawing a specific number of successes from a finite population without replacement. It is fundamental when samples are drawn without replacement from a known population that contains a fixed number of items of different types, and it is used to calculate probabilities using combinatorial counts rather than assuming independent trials.

Transcript

00:01 Hi guys, it's given that the website contains 100 pages and the 60, 30 and 10 % of the page contain low, moderate and high graphic content respectively.

00:14 So here we have 60 % contain low and 30 % contains moderate and 10 % contain high graphic content.

00:35 Okay, so a sample of 4 page is thrown without replacement.

00:39 So let's, um, x and y are the random variables that denote the number of page with moderate and high with moderate high graphic output in the sample.

00:51 So the moderate is x and the high is y.

00:55 Okay.

00:57 So the objective is to find f of x and y and we know that f of x and y equals.

01:10 30 cx times 10 c y times 60 times c4 minus x over 100 c4 okay so if you construct a table for 4 values of x and y okay so let's construct it we have six columns i'm sorry okay so here is x or y then 0 1 2 3 or 4 and a total okay so here is x and here's y 0 1 2 3 4 and total okay so it's also here's 0 1 2 3 4 and total okay so it's also here's 0 1 2 2 2 2 2 i'm sorry you guys, so 2, 3, 4 and the total.

03:21 So 3, 4, and here we have the total.

03:26 Okay, so let's start the probability of 0 and 0, so it's 0 .144.

03:34 Okay, and here is 0 .0873, and here is 0 .0203, and here is 0 .02031 .3 .1 .0 .31 .1 .1.

03:45 There is 0 .018 there is 0 .001 and the total is 0 .234.

03:58 Okay then it's 0 .2618 then it's 0 .13542 then it's 0 .13542 then it's 0 .0 .0266.

04:16 Okay then it's 0 .0092 then 0 okay so it's 0 .419 okay then it's all point 19 okay then it's all point 1964 then it's 0 .0666 then it's 0 .00 then it's 0 .04 so it's 0 .268.

04:53 Okay.

04:55 Then here we have all point.

04:59 Yes, okay.

05:02 Here we have 0 .06 to 1...

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