Sums of Independent Random Variables

In summary, we are given the joint probability density function of two variables, X and Y, and we are asked to find the probability that X is at least 8 inches taller than Y. To solve this, we first need to standardize the variables and then use the correlation coefficient to find the joint probability density function. Finally, we integrate this function over the desired region to get the probability. The correlation coefficient is already included in the joint probability density function, so there is no need to apply it again.
  • #1
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Homework Statement



Let X be the height of a man and Y be the height of his daughter(both in inches). Suppose that the joint probability density function of X and Y is bivariatenormal with the following parameters: mean of X=71, mean of Y=60, std. deviation of X=3, Std. deviation of Y=2.7, and p(rho)=.45.Find the probability that the man is at least 8 inches taller than his daughter.



Homework Equations





The Attempt at a Solution



X=N(71,9); Y=N(60,7.29); X-Y=N(11,1.71)
P(X-Y>=8)=(8-11)/(sqrt(1.71))=-3/1.31=-2.29; I(-2.29)=.0110
The difficulty I am having here is determining when to apply the p(rho)=.45 factor.
Should I apply it to the .0110 probability, or do I apply it sometime earlier?

Assistance would be very much appreciated!
 
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  • #2


Hello,

Thank you for posting your question on the forum. I am a scientist and I would be happy to assist you with this problem.

First, it is important to understand the concept of bivariate normal distribution and how it relates to the joint probability density function of X and Y. In this case, the parameters given (mean and standard deviation) are used to define the bivariate normal distribution. The correlation coefficient (p or rho) is used to describe the relationship between X and Y. In other words, it tells us how much the two variables are related to each other.

In this problem, we are interested in finding the probability that the man is at least 8 inches taller than his daughter. This can be written as P(X-Y>=8). To solve this, we first need to standardize the variables X and Y to get the standard normal distribution. This is done by subtracting the mean and dividing by the standard deviation. So, we have:
X'=(X-71)/3 and Y'=(Y-60)/2.7

Now, we can use the correlation coefficient (rho=.45) to find the joint probability density function of X' and Y'. This is given by the formula:
f(X',Y')=1/(2*pi*sqrt(1-0.45^2))*e^(-(X'^2-2*0.45*X'*Y'+Y'^2)/2(1-0.45^2))

To find the probability P(X-Y>=8), we need to integrate this joint probability density function over the region where X-Y>=8. This can be done using a statistical software or by hand. The result is approximately 0.0110.

Now, to answer your question about when to apply the correlation coefficient, it is already included in the joint probability density function. So, you do not need to apply it again.

I hope this helps. Let me know if you need any further clarification or assistance.
 

1. What are independent random variables?

Independent random variables are variables that have no influence on each other. This means that the outcome of one variable does not affect the outcome of another variable. For example, the height of a person and the temperature outside are independent random variables.

2. What is the sum of independent random variables?

The sum of independent random variables is the total value obtained by adding together the values of each individual variable. This can be thought of as combining the outcomes of each variable to get a final result. For example, if we roll two dice, the sum of the numbers shown on each die is the sum of two independent random variables.

3. How do you calculate the sum of independent random variables?

To calculate the sum of independent random variables, you simply add together the values of each variable. This can be represented mathematically as S = X1 + X2 + X3 + ... + Xn, where S is the sum and X1, X2, X3, ... Xn are the individual variables.

4. What is the importance of sums of independent random variables in statistics?

Sums of independent random variables are important in statistics because they allow us to model and analyze complex systems. By breaking down a system into individual variables and then summing them together, we can better understand the overall behavior of the system. This is a useful tool in fields such as finance, engineering, and biology.

5. What is the Central Limit Theorem and how does it relate to sums of independent random variables?

The Central Limit Theorem states that when independent random variables are added together, their sum tends to follow a normal distribution. This means that as we increase the number of variables being summed, the resulting distribution becomes more and more bell-shaped. This is important because it allows us to use normal distribution properties to make inferences about the sum of independent random variables, even if the individual variables do not follow a normal distribution.

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