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Chapter 3: Problem 2

Let \(X\) and \(Y\) have a bivariate normal distribution with parameters \(\mu_{1}=\) 3, \(\mu_{2}=1, \sigma_{1}^{2}=16, \sigma_{2}^{2}=25\), and \(\rho=\frac{3}{5} .\) Determine the following probabilities: (a) \(P(3

Short Answer

Expert verified
The exact answer to these probabilities depends on the Z-table used or calculator accuracy. However, the above steps will guide on how to approach each part of the problem.

Step by step solution

01

Understand the given parameters

The parameters for the bivariate normal distribution are given as follow: mean of X \(\mu_{1}=3\), mean of Y \(\mu_{2}=1\), variance of X \(\sigma_{1}^{2}=16\), variance of Y \(\sigma_{2}^{2}=25\) and correlation coefficient \(\rho=\frac{3}{5}\). These will be used to calculate the probabilities.
02

Calculate the probabilities

To solve these problems, use the cumulative distribution function (CDF) of the normal distribution, which in practice means using statistical tables or a calculator. (a) \(P(3<Y<8)\) means we are looking for the probability that Y takes a value between 3 and 8. We could standardize the variables by subtracting the mean and dividing by the standard deviation, then use the look-up table for the standard normal distribution. (b) \(P(3<Y<8|X=7)\) represents a conditional probability. We can calculate this using the properties of bivariate normal distribution where mean and variance transform given the condition. (c) \(P(-3<X<3)\) and (d) \(P(-3<X<3|Y=-4)\) can be calculated similarly to parts (a) and (b) respectively. However, they relate to variable X.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conditional Probability
In simple terms, conditional probability defines the likelihood of an event or outcome occurring, given the occurrence of another event. In the context of the exercise, we're examining the bivariate normal distribution, which is a type of probability distribution for two correlated random variables. When we talk about conditional probabilities with respect to bivariate distributions, it means we are interested in the probability of one variable falling within a certain range given that another variable is fixed at a specific value.

For example, in the exercise, part (b) \(P(3
Cumulative Distribution Function (CDF)
The cumulative distribution function (CDF) is a fundamental concept in statistics. It gives the probability that a random variable is less than or equal to a certain value. Specifically, for any number \(x\), the CDF \(F(x)\) of a random variable \(X\) is the probability that \(X \leq x\).

In our exercise, we utilize the CDF to calculate probabilities over ranges (e.g. \(P(3
Standard Normal Distribution
The standard normal distribution is a special case of the normal distribution. It's defined as a normal distribution with a mean (\(\mu\)) of 0 and a variance (\(\sigma^2\)) of 1. Given its standardized nature, it provides a reference framework to calculate probabilities for any normally distributed variable once it has been standardized, meaning we subtract the mean and divide by the standard deviation.

In the context of our exercise, we standardize when we use a z-table to find probabilities. For instance, to find \(P(-3

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Most popular questions from this chapter

. Let \(X\) and \(Y\) have a bivariate normal distribution with parameters \(\mu_{1}=\) \(\mu_{2}=0, \sigma_{1}^{2}=\sigma_{2}^{2}=1\), and correlation coefficient \(\rho .\) Find the distribution of the random variable \(Z=a X+b Y\) in which \(a\) and \(b\) are nonzero constants.

Let \(f(x)\) and \(F(x)\) be the pdf and the cdf of a distribution of the continuous type such that \(f^{\prime}(x)\) exists for all \(x\). Let the mean of the truncated distribution that has pdf \(g(y)=f(y) / F(b),-\infty

. Let \(X\) have a conditional Burr distribution with fixed parameters \(\beta\) and \(\tau\), given parameter \(\alpha\). (a) If \(\alpha\) has the geometric pdf \(p(1-p)^{\alpha}, \alpha=0,1,2, \ldots\), show that the unconditional distribution of \(X\) is a Burr distribution. (b) If \(\alpha\) has the exponential pdf \(\beta^{-1} e^{-\alpha / \beta}, \alpha>0\), find the unconditional pdf of \(X\).

Let the number of chocolate drops in a certain type of cookie have a Poisson distribution. We want the probability that a cookie of this type contains at least two chocolate drops to be greater than \(0.99 .\) Find the smallest value of the mean that the distribution can take.

If \(X\) is \(N(75,100)\), find \(P(X<60)\) and \(P(70
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