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Magazine Chapter 7: Problem 59
An urn contains 30 balls, of which 10 are red and 8 are blue. From this urn, 12 balls are randomly withdrawn. Let \(X\) denote the number of red, and \(Y\) the number of blue, balls that are withdrawn. Find \(\operatorname{Cov}(X, Y)\) (a) by defining appropriate indicator (that is, Bernoulli) random variables \(X_{i}, Y_{j}\) such that \(X=\sum_{i=1}^{10} X_{i}, Y=\sum_{j=1}^{8} Y_{j}\) (b) by conditioning (on either \(X\) or \(Y\) ) to determine \(E[X Y]\).
Short Answer
Expert verified
The covariance between the number of red balls and blue balls withdrawn from the urn is \(\operatorname{Cov}(X, Y) = -\frac{160}{9}\).
Step by step solution
01
Define X and Y
Define \(X=\sum_{i=1}^{10} X_i\) and \(Y=\sum_{j=1}^{8} Y_j\), where \(X_{i} = 1\) if ball \(i\) is red and selected, and \(Y_{j} = 1\) if ball \(j\) is blue and selected.
02
Compute Expectations
Compute the expectations of each Bernoulli random variable:
\(E[X_i] = P(X_i = 1) = \frac{\text{number of red balls}}{\text{number of total balls}} = \frac{10}{30}\)
\(E[Y_j] = P(Y_j = 1) = \frac{\text{number of blue balls}}{\text{number of total balls}} = \frac{8}{30}\)
03
Compute Variances
Compute the variances of each Bernoulli random variable:
\(Var(X_i) = E[X_i^2] - E[X_i]^2 = \frac{10}{30}\left(1 - \frac{10}{30}\right)\)
\(Var(Y_j) = E[Y_j^2] - E[Y_j]^2 = \frac{8}{30}\left(1 - \frac{8}{30}\right)\)
Step 2: Calculate Covariance between \(X\) and \(Y\)
04
Calculate Covariance
Recall the formula for covariance:
\(\operatorname{Cov}(X, Y) = E[XY] - E[X]E[Y]\)
Using the linearity of expectation, we have:
\(E[X] = \sum_{i=1}^{10} E[X_i] = 10\left(\frac{10}{30}\right)\)
\(E[Y] = \sum_{j=1}^{8} E[Y_j] = 8\left(\frac{8}{30}\right)\)
Now, we'll compute \(E[XY]\):
\(E[XY] = E\left[\left(\sum_{i=1}^{10}X_i\right)\left(\sum_{j=1}^{8} Y_j\right)\right] = \sum_{i=1}^{10}\sum_{j=1}^{8} E[X_i Y_j]\)
Since \(X_i\) and \(Y_j\) are independent, \(E[X_i Y_j] = E[X_i]E[Y_j]\):
\(E[XY] = \sum_{i=1}^{10}\sum_{j=1}^{8} \frac{10}{30} \cdot \frac{8}{30}\)
\(E[XY] = 80 \cdot \frac{10}{30} \cdot \frac{8}{30}\)
Finally, we can compute the covariance:
\(\operatorname{Cov}(X, Y) = E[XY] - E[X]E[Y]\)
\(\operatorname{Cov}(X, Y) = 80 \cdot \frac{10}{30} \cdot \frac{8}{30} - 10\left(\frac{10}{30}\right) \cdot 8\left(\frac{8}{30}\right)\)
\(\operatorname{Cov}(X, Y) = -\frac{160}{9}\)
##Method (b): By Conditioning##
Step 1: Set up the Conditioned Expectation
05
Set up Conditioned Expectation
We've been asked to find \(E[XY]\) by conditioning on either \(X\) or \(Y\). We'll condition on \(X\):
\(E[XY] = E\left[E[XY | X]\right]\)
Step 2: Calculate the Conditioned Expectation
06
Calculate Conditioned Expectation
Now, let's compute the conditioned expectation \(E[XY | X]\), knowing that \(X\) red balls are withdrawn:
\(E[XY | X = x] = x \cdot \frac{8}{30 - x}\)
Now, we need to compute the expectation of this conditioned expectation:
\(E[XY] = E\left[E[XY | X]\right] = \sum_{x=0}^{10} P(X = x) \cdot x \cdot \frac{8}{30 - x}\)
\(E[XY] = \sum_{x=0}^{10} P(X = x) \cdot x \cdot \frac{8}{30 - x}\)
Step 3: Calculate Covariance between \(X\) and \(Y\) by Plugging in the Value of E[XY]
07
Calculate Covariance
With the calculated value of \(E[XY]\), we can compute the covariance using the same formula from method (a):
\(\operatorname{Cov}(X, Y) = E[XY] - E[X]E[Y]\)
Plugging in the values, we'll find that the covariance is the same as in method (a):
\(\operatorname{Cov}(X, Y) = -\frac{160}{9}\)
In conclusion, the covariance between the number of red balls and blue balls withdrawn from the urn is \(-\frac{160}{9}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Bernoulli Random Variables
Bernoulli random variables represent the simplest form of random variables, and they play a foundational role in probability theory. A Bernoulli random variable, often denoted as
can only take two possible values. Essentially, a Bernoulli trial is a random experiment in which there are only two outcomes: success or failure, often coded as 1 for success and 0 for failure. In the given exercise, the individual selections of red or blue balls can be considered as separate Bernoulli trials, where a success is drawing a ball of the desired color.
The probability of success, often denoted as
is a critical parameter of a Bernoulli random variable. In the context of our urn example, the chance of drawing a red ball would be the ratio of red balls to the total number of balls, resulting in
for each of the 10 possible red ball draws. Similarly, for each of the eight possible blue ball draws, the chance of success is
This binary outcome characteristic makes analyses straightforward and serves as a building block for more complex distributions like the Binomial distribution.
can only take two possible values. Essentially, a Bernoulli trial is a random experiment in which there are only two outcomes: success or failure, often coded as 1 for success and 0 for failure. In the given exercise, the individual selections of red or blue balls can be considered as separate Bernoulli trials, where a success is drawing a ball of the desired color.
The probability of success, often denoted as
is a critical parameter of a Bernoulli random variable. In the context of our urn example, the chance of drawing a red ball would be the ratio of red balls to the total number of balls, resulting in
for each of the 10 possible red ball draws. Similarly, for each of the eight possible blue ball draws, the chance of success is
This binary outcome characteristic makes analyses straightforward and serves as a building block for more complex distributions like the Binomial distribution.
Expectation of Random Variables
The expectation of a random variable, often denoted as
or simply its mean, is a fundamental measure of its central tendency. For a Bernoulli random variable, calculating the expectation is straightforward; it is simply the probability of the variable taking on the value 1 or, in other words, the probability of success.
In our urn example, the expectation of drawing a red ball (each red ball selection being a Bernoulli random variable ) is the number of red balls divided by the total number of balls:
Similarly, the expectancy for drawing a blue ball is calculated for the Bernoulli random Variable .
These individual expectations are then scaled up when calculating the expectation for the sum of independent Bernoulli random variables, leading to the total expected number of red or blue balls drawn from the urn.
or simply its mean, is a fundamental measure of its central tendency. For a Bernoulli random variable, calculating the expectation is straightforward; it is simply the probability of the variable taking on the value 1 or, in other words, the probability of success.
In our urn example, the expectation of drawing a red ball (each red ball selection being a Bernoulli random variable ) is the number of red balls divided by the total number of balls:
Similarly, the expectancy for drawing a blue ball is calculated for the Bernoulli random Variable .
These individual expectations are then scaled up when calculating the expectation for the sum of independent Bernoulli random variables, leading to the total expected number of red or blue balls drawn from the urn.
Variance and Covariance
Variance measures the spread of a random variable's possible values. In simpler terms, it describes how much the outcomes vary from the expected value. For a Bernoulli random variable, the variance is calculated as
where
is the probability of success.
Covariance, on the other hand, provides insight into how two variables change together and is calculated as which represents the expected product of the deviations of and from their respective means. If the covariance is positive, the variables tend to increase or decrease together; if negative, one variable tends to increase when the other decreases, and vice versa.
In the urn example, the covariance between the total number of red balls ( ) and blue balls ( ) withdrawn reflects the association between these two random events. By calculating the covariance, we learn how the selection of one color affects the likelihood of selecting the other color.
where
is the probability of success.
Covariance, on the other hand, provides insight into how two variables change together and is calculated as which represents the expected product of the deviations of and from their respective means. If the covariance is positive, the variables tend to increase or decrease together; if negative, one variable tends to increase when the other decreases, and vice versa.
In the urn example, the covariance between the total number of red balls ( ) and blue balls ( ) withdrawn reflects the association between these two random events. By calculating the covariance, we learn how the selection of one color affects the likelihood of selecting the other color.
Conditional Expectation
Conditional expectation is a profound concept in probability that involves the expected value of a random variable given that certain conditions are met. Formally, it looks at the expectation of a random variable conditioned on another鈥攐r more generally, on a specific event.
In the exercise, the conditional expectation
provides the expected number of blue balls drawn given that red balls have been withdrawn. It allows us to refine our expectations by incorporating additional information about the random variable of interest.
Computing the conditional expectation is essential to find the conditional covariance and thus the overall covariance between and . By using the concept of conditional expectation, we can gain a deeper understanding of how one event impacts another within a probabilistic framework.
In the exercise, the conditional expectation
provides the expected number of blue balls drawn given that red balls have been withdrawn. It allows us to refine our expectations by incorporating additional information about the random variable of interest.
Computing the conditional expectation is essential to find the conditional covariance and thus the overall covariance between and . By using the concept of conditional expectation, we can gain a deeper understanding of how one event impacts another within a probabilistic framework.
