Question

Let \(X\) and \(Y\) be independent random variables, each with a distribution that is \(N(0,1) .\) Let \(Z=X+Y .\) Find the integral that represents the cdf \(G(z)=\) \(P(X+Y \leq z)\) of \(Z .\) Determine the pdf of \(Z\). Hint: We have that \(G(z)=\int_{-\infty}^{\infty} H(x, z) d x\), where $$ H(x, z)=\int_{-\infty}^{z-x} \frac{1}{2 \pi} \exp \left[-\left(x^{2}+y^{2}\right) / 2\right] d y $$ Find \(G^{\prime}(z)\) by evaluating \(\int_{-\infty}^{\infty}[\partial H(x, z) / \partial z] d x\).

Short answer

The integral representing the cumulative distribution function (CDF) G(z) is given by \( G(z)=\int_{-\infty}^{\infty}\int_{-\infty}^{z-x} \frac{1}{2 \pi} \exp[-(x^{2}+y^{2}) / 2] dy dx \). And after differentiating the CDF with respect to z we get the probability density function (PDF) of Z, which is \( G'(z)=\frac{1}{\sqrt{2\pi}} \exp \left(-\frac{z^{2}}{4}\right) \).

Step-by-step solution

Understand the given data

We have two independent random variable X and Y, both with a standard normal distribution (\(N(0,1)\)). A third variable Z is defined as the sum of X and Y, \(Z=X+Y\). We need to find the cumulative distribution function (CDF), \(G(z)\), and the probability density function (PDF) of Z.

Find the integral representing the CDF

Using the hint, \( G(z)=\int_{-\infty}^{\infty} H(x, z) dx \), where \( H(x,z)=\int_{-\infty}^{z-x} \frac{1}{2 \pi} \exp \left[-\left(x^{2}+y^{2}\right) / 2\right] dy \). This serves as the CDF of Z.

Determine the derivative of H with respect to z

This is the key step to finding the PDF of Z. Replace H(x,z) in the derivative using the above information and find:\[ \frac{\partial H(x,z)}{\partial z} = \frac{1}{2 \pi} \exp \left[-\left((z-x)^{2}+x^{2}\right) / 2\right] \]

Find the PDF by differentiating the CDF

Since \(G(z)\) represents the CDF of Z, the PDF of Z can be obtained by differentiating the CDF \(G(z)\) with respect to z. Perform the integration to get \(G'(z)\):\[G'(z) = \int_{-\infty}^{\infty} \frac{\partial H(x,z)}{\partial z} dx = \int_{-\infty}^{\infty} \frac{1}{2 \pi} \exp \left[-\left((z-x)^{2}+x^{2}\right) / 2\right] dx \]After simplifying, we find that the result is \( G'(z)=\frac{1}{\sqrt{2\pi}} \exp \left(-\frac{z^{2}}{4}\right) \)

Key concepts

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) is a crucial concept in the study of probability. For a given random variable, the CDF represents the probability that this variable will take on a value less than or equal to a specific value. In simpler terms, it tells us how much of the probability lies to the left of a particular point on the probability distribution's graph.

For the standard normal distribution, denoted by \(N(0,1)\), the CDF is a bell-shaped curve that indicates the probability that a variable will fall within a particular range. It’s important to recognize that the area under the entire curve is equal to 1, which signifies a complete certainty that any value of the variable will fall within the bounds of the distribution.

In the exercise, we were asked to determine the CDF of a new variable \(Z\), which is the sum of two independent standard normal random variables. The calculation involved an integration process, which demonstrates how CDFs can help us find probabilities for more complex, composite variables.

Standard Normal Distribution

The standard normal distribution, often symbolized as \(N(0,1)\), is a special case of the normal distribution that has a mean (\(\mu\)) of 0 and a standard deviation (\(\sigma\)) of 1. This distribution is fundamental in statistics because it serves as the benchmark for measuring how data deviate from the mean, which is especially useful for hypothesis testing and creating confidence intervals.

When working with the standard normal distribution, one often refers to the Z-score, which represents the number of standard deviations away from the mean a data point is. The process of transforming a random variable into a Z-score is known as standardization, which enables us to make comparisons between data from different normal distributions. The CDF of the standard normal distribution can be denoted by \(\Phi(z)\), and in the context of our exercise, knowing this function would help us to calculate probabilities directly using standard normal tables or software.

Independent Random Variables

Independent random variables are a fundamental concept in probability theory. Two variables are independent if the realization of one does not affect the probability distribution of the other. In mathematical terms, \(X\) and \(Y\) are independent if the probability of them occurring together is the product of their individual probabilities: \(P(X = x \text{ and } Y = y) = P(X = x)P(Y = y)\).

In our exercise, \(X\) and \(Y\) are independent standard normal random variables, which implies that knowing the value of \(X\) gives no information about \(Y\) and vice versa. This independence is vital for simplifying the computation of the CDF and PDF of the sum of these variables, \(Z\). If \(X\) and \(Y\) were not independent, the computations would involve a more complex, joint probability distribution.