Problem 71 - An Introduction to Mathematical Statistics and Its Applications

Question

Suppose that \(X\) and \(Y\) have a bivariate normal pdf with \(\mu_{X}=3, \mu_{Y}=6, \sigma_{X}^{2}=4, \sigma_{Y}^{2}=10\), and \(\rho=\frac{1}{2}\). Find \(P\left(5

Step-by-Step Solution

Answer

Therefore, the values of \( P\left(5

Step 1: Understanding the Parameters

Given the parameters of the bivariate normal distribution, recognize that \( \mu_{X}=3 \), \( \mu_{Y}=6 \), \( \sigma_{X}^{2}=4 \), \( \sigma_{Y}^{2}=10 \), and \( \rho=\frac{1}{2} \). These are the mean of X and Y (which are $\mu_{X}$ and $\mu_{Y}$), the variance of X and Y (which are \( \sigma_{X}^{2} \) and \( \sigma_{Y}^{2} \)), and the correlation coefficient \( \rho \) which measures the degree of relationship between X and Y.

Step 2: Find the Probability \( P\left(5<Y<6 \frac{1}{2}\right) \)

Using cumulative distribution function with upper limit \( 6 \frac{1}{2} \) and lower limit 5, we can find the Probability \[ P\left(5

Step 3: Find conditional variance

Next, we find the conditional variance of Y given X since that's needed find \( P\left(5

Step 4: Find conditional mean

The next step is to find conditional mean of Y given X. This is calculated by: \[ \mu_{Y|X} = \mu_{Y} + \rho \left( x-\mu_{X} \right) \frac{\sigma_{Y}}{\sigma_{X}} = 6 + 0.5 \times (2 - 3) \times \frac{\sqrt{10}}{2} = 5.5 \] Here \( \mu_{Y|X} \) is the conditional mean of Y given X is 2.

Step 5: Find \( P\left(5<Y<6 \frac{1}{2} \mid x=2\right) \)

Finally, we can calculate \( P\left(5

Key Concepts

Probability Density Function

Imagine you're sitting on a bench in a park, and you want to predict where the next person will sit down. This is akin to understanding how the probability density function (pdf) helps describe the likelihood of a random variable, like where a person chooses to sit. In the realm of the bivariate normal distribution, the pdf serves as a mathematical model that describes the probability of two continuous random variables occurring together.

In our exercise, we have two related variables, X and Y, each with their own average sitting spots (mean values) and preferred spaces (variances). The bivariate normal pdf is the joint pdf of these two variables, factoring in not just where they like to sit alone (univariate distributions), but also how their seating preferences affect one another (their correlation). The higher the pdf at a certain point, the more likely you are to observe the two friends sitting in that spot in the park. The pdf, therefore, provides a foundation for calculating probabilities for both individual and related events concerning X and Y.

Conditional Probability

Now, let's say we want to offer a free cup of coffee to the next person who sits down, but we're only interested if that person arrives with a friend. We're now thinking about conditional probability, which in statistical terms, is the likelihood of an event occurring given that another event has already happened.

In our exercise, we're specifically looking at the probability that Y falls between two values, given that X has a specific value. Imagine that X is a usual seating spot for a friend, and we want to know the chances that another friend, Y, will sit nearby. The solution requires adjusting our understanding of Y's behavior (its distribution) based on where X is seated. By calculating conditional means and variances, we refine our probabilities to account for this relationship, giving us a more accurate idea of where to expect our coffee-drinking friend Y to sit down.

Cumulative Distribution Function

Consider a contest where the first person to sit between two particular benches wins a prize. The cumulative distribution function (CDF) can predict the likelihood of this happening. It represents the probability that a variable will be found at a value less than or equal to a certain point. So what are the odds someone will have won the prize as you arrive? This is the kind of question the CDF can answer.

In our pair of seated friends X and Y, we can use the CDF to find the probability that Y sits down within a certain range of our park bench. By determining the CDF for the upper and lower limits of this range and subtracting the two, we can predict the chances of our successful seat selection. The CDF steps in our solution effectively tell us how likely it is for a person to sit within our winning seats before we even lay out the prize.

Variance and Mean

If we think of our park seats as having particular favorite seating spots and variabilities in sitting preference, that's similar to the way variance and mean operate in statistics. The mean provides the average or expected value, akin to a favored seat in the park. While the variance tells us how spread out people's seating preferences are from that favorite spot - the larger the variance, the less predictable the seating pattern.

In our bivariate example, we're given the mean and variance for both individuals X and Y. These statistics are essential for setting the parameters of our pdf and calculating both direct and conditional probabilities. When X picks a seat (a specific value), we adjust Y's mean and variance for that condition, giving us a more nuanced understanding of where Y might choose to sit.

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Problem 69

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Problem 72

Other exercises in this chapter

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Problem 72

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Problem 73

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Key Concepts

Probability Density Function

Imagine you're sitting on a bench in a park, and you want to predict where the next person will sit down. This is akin to understanding how the probability density function (pdf) helps describe the likelihood of a random variable, like where a person chooses to sit. In the realm of the bivariate normal distribution, the pdf serves as a mathematical model that describes the probability of two continuous random variables occurring together.

In our exercise, we have two related variables, X and Y, each with their own average sitting spots (mean values) and preferred spaces (variances). The bivariate normal pdf is the joint pdf of these two variables, factoring in not just where they like to sit alone (univariate distributions), but also how their seating preferences affect one another (their correlation). The higher the pdf at a certain point, the more likely you are to observe the two friends sitting in that spot in the park. The pdf, therefore, provides a foundation for calculating probabilities for both individual and related events concerning X and Y.

Conditional Probability

Now, let's say we want to offer a free cup of coffee to the next person who sits down, but we're only interested if that person arrives with a friend. We're now thinking about conditional probability, which in statistical terms, is the likelihood of an event occurring given that another event has already happened.

In our exercise, we're specifically looking at the probability that Y falls between two values, given that X has a specific value. Imagine that X is a usual seating spot for a friend, and we want to know the chances that another friend, Y, will sit nearby. The solution requires adjusting our understanding of Y's behavior (its distribution) based on where X is seated. By calculating conditional means and variances, we refine our probabilities to account for this relationship, giving us a more accurate idea of where to expect our coffee-drinking friend Y to sit down.

Cumulative Distribution Function

Consider a contest where the first person to sit between two particular benches wins a prize. The cumulative distribution function (CDF) can predict the likelihood of this happening. It represents the probability that a variable will be found at a value less than or equal to a certain point. So what are the odds someone will have won the prize as you arrive? This is the kind of question the CDF can answer.

In our pair of seated friends X and Y, we can use the CDF to find the probability that Y sits down within a certain range of our park bench. By determining the CDF for the upper and lower limits of this range and subtracting the two, we can predict the chances of our successful seat selection. The CDF steps in our solution effectively tell us how likely it is for a person to sit within our winning seats before we even lay out the prize.

Variance and Mean

If we think of our park seats as having particular favorite seating spots and variabilities in sitting preference, that's similar to the way variance and mean operate in statistics. The mean provides the average or expected value, akin to a favored seat in the park. While the variance tells us how spread out people's seating preferences are from that favorite spot - the larger the variance, the less predictable the seating pattern.

In our bivariate example, we're given the mean and variance for both individuals X and Y. These statistics are essential for setting the parameters of our pdf and calculating both direct and conditional probabilities. When X picks a seat (a specific value), we adjust Y's mean and variance for that condition, giving us a more nuanced understanding of where Y might choose to sit.