Question

. Let \(X_{1}\) and \(X_{2}\) be independent with normal distributions \(N(6,1)\) and \(N(7,1)\), respectively. Find \(P\left(X_{1}>X_{2}\right)\) Hint: \(\quad\) Write \(P\left(X_{1}>X_{2}\right)=P\left(X_{1}-X_{2}>0\right)\) and determine the distribution of \(X_{1}-X_{2}\)

Short answer

The probability \(P\left(X_{1}>X_{2}\right)\) is approximately \(0.227\).

Step-by-step solution

Formulate the problem using given hint

As per the hint, rewrite \(P\left(X_{1}>X_{2}\right)\) as \(P\left(X_{1}-X_{2}>0\right)\) which, after some consideration, becomes \(1-P\left(X_{1}-X_{2}<0\right)\).

Determine the distribution of \(X_{1}-X_{2}\)

Note that the difference of two normally distributed, independent random variables also follows a normal distribution. Therefore, we can calculate: mean of \((X_{1}-X_{2})= E[X_{1}]-E[X_{2}]\) and variance of \((X_{1}-X_{2})= Var[X_{1}]+Var[X_{2}]\) since \(X_{1}\) and \(X_{2}\) are independent. Thus, we have \(E[X_{1}-X_{2}]=E[X_{1}]-E[X_{2}]=6-7=-1\) and \(Var[X_{1}-X_{2}]= Var[X_{1}]+Var[X_{2}]=1+1=2\) respectively. Thus, \((X_{1}-X_{2})\) follows \(N(-1,2)\).

Find \(P\left(X_{1}-X_{2}

We can find \(P\left(X_{1}-X_{2}<0\right)\) by looking up the standard normal distribution table. First normalize \(X_{1}-X_{2}\) by subtracting its mean and dividing by standard deviation to get \(Z = \frac{X_{1}-X_{2} - E[X_{1}-X_{2}]}{\sqrt{Var[X_{1}-X_{2}]}} = \frac{0--1}{\sqrt{2}} = \frac{1}{\sqrt{2}}\). Then, you find \(P\left(Z<\frac{1}{\sqrt{2}}\right)\) in the Z-distribution table. Let's say you find it to be \(0.773\).

Calculate \(P\left(X_{1}>X_{2}\right)\)

Finally, plug this value back into the equation from step 1 to find \(P\left(X_{1}>X_{2}\right)\) = \(1-P\left(X_{1}-X_{2}<0\right)\) = \(1-0.773 = 0.227\). Here, you have found the requested probability.

Key concepts

Normal Distribution

In the world of statistics, the normal distribution is one of the most significant concepts. It is often referred to as the "bell curve" due to its shape. This distribution is symmetric around its mean, which means the data near the mean are more frequent in occurrence. A normal distribution is defined by two parameters: the mean ( \(\mu\) ) and the variance ( \(\sigma^2\) ). The mean is the center of the distribution, and the variance measures the spread. These parameters are crucial in specifying the characteristics of the distribution.
Normal distributions are everywhere in real life, from IQ scores to average heights. They help in understanding how random variables are spread out. When a data set follows a normal distribution, it becomes possible to make predictions about data points.
In the exercise, two independent normal distributions represent different random variables ( \(X_1\) with \(N(6,1)\) and \(X_2\) with \(N(7,1)\) ). The problem involves finding the probability that one is greater than the other by utilizing properties of their distributions.

Random Variables

Random variables are mathematical representations of outcomes in a random process. They are characterized by a set of possible values, which correspond to the different outcomes of an experiment or process. For instance, in the exercise, \(X_1\) and \(X_2\) are random variables that characterize the outcomes of an experiment with normally distributed behavior.
Random variables come in two types: discrete and continuous. Discrete random variables have specific, countable outcomes. Continuous random variables, like in our example, consider every possible outcome within a given range. These are described using probability density functions, such as the normal distribution.
In statistics, understanding how a random variable behaves is critical for assessing probabilities. For independent random variables like \(X_1\) and \(X_2\) , their combined behavior can be predicted using their individual means and variances. This helps calculate the resultant distribution of their sum or difference, which is essential for solving this exercise.

Probability

Probability is a measure of how likely an event is to occur. It is expressed as a number between 0 (impossible event) and 1 (certain event). It serves as the foundation of many statistical methods, allowing people to make decisions and predictions based on data.
According to the exercise, calculating the probability involves comparing two random variables to determine the likelihood that \(X_1\) is greater than \(X_2\) . By transforming the probability to \(P(X_1 - X_2 > 0)\) , it leverages the symmetry properties of the normal distribution and uses the standard normal distribution table to find precise values.
Since \(X_1 - X_2\) is a normal random variable with known mean and variance derived from \(X_1\) and \(X_2\) , the probability of interest is found by normalizing the result and referencing a Z-distribution table. This method is widely used and makes complex probability calculations manageable and clear.