Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a continuous-type distribution. (a) Find \(P\left(X_{1} \leq X_{2}\right), P\left(X_{1} \leq X_{2}, X_{1} \leq X_{3}\right), \ldots, P\left(X_{1} \leq X_{i}, i=2,3, \ldots, n\right)\) (b) Suppose the sampling continues until \(X_{1}\) is no longer the smallest observation (i.e., \(X_{j}

Short answer

For the given continuous-type distribution: (a) \(P\left(X_{1} \leq X_{2}\right) = P\left(X_{1} \leq X_{3}\right) = \ldots = P\left(X_{1} \leq X_{n}\right) = 0.5\), (b) \(P(Y=y)=\frac{1}{y(y+1)}, y=1,2,3, \ldots\), and (c) Neither the mean nor the variance of \(Y\) exist.

Step-by-step solution

Determine the probability for each \(X_{i}\)

For random variables from a continuous-type distribution, the probability that one variable is less than or equal to another is always 0.5, assuming the variables are independent and follow the same distribution. Thus, \(P\left(X_{1} \leq X_{2}\right) = P\left(X_{1} \leq X_{3}\right) = \ldots = P\left(X_{1} \leq X_{n}\right) = 0.5.\)

Determine the general form of the probability distribution of \(Y\)

To derive the general form of the probability that \(Y = y\), consider the situation in which \(Y\) is the number of trials before \(X_{1}\) is no longer the smallest observation. This problem can be equivalent to arranging \(y+1\) numbers, where one is \(X_{1}\) and the others are larger than \(X_{1}\). Because every order is equally likely, the probability we are interested in is one divided by the total number of possible orders, which is \(y+1\). Since we are only interested in cases where \(X_{1}\) is not the largest number, we subtract one possible order, resulting in \(y\). Thus, \(P(Y=y)=\frac{1}{y(y+1)}, \quad y=1,2,3, \ldots\)

Compute the mean and variance of \(Y\)

First, the mean or expected value of \(Y\) can be calculated by summing over all the possible values that \(Y\) can take, multiplied by their respective probabilities. However, as \(y\) goes from 1 to infinity, the series does not converge, indicating the mean does not exist. \nTo determine if the variance exists, we need to calculate the second moment \(E[Y^2]\) firstly using the formula \(E[Y^2]=\sum_{y=1}^{\infty} y^2P(Y=y)\). If this series converges, then the variance will be \(Var(Y) = E[Y^2] - (E[Y])^2\), given that \(E[Y]\) exists. However, both series for \(E[Y]\) and \(E[Y^2]\) are infinite, indicating that neither the mean nor the variance of \(Y\) exist.

Key concepts

Random Sample

When discussing statistics, a random sample is a fundamental concept that refers to a set of observations extracted from a population, where each member of the population has an equal chance of being included. The importance of randomness is to ensure that the sample represents the larger group without bias.

For example, if you want to know the average height of students in a school, randomly selecting a group of students to measure will likely give you an accurate representation. This process is critical in statistical analysis because it underpins the validity of your conclusions. If the sample isn't random, the results could be skewed due to sampling bias, which would make the findings unreliable.

Continuous-Type Distribution

In probability theory, a continuous-type distribution describes a random variable that can take on any value within a certain range. These distributions are characterized by having a continuous set of possible outcomes, unlike discrete distributions which have countable outcomes.

A classic example of a continuous distribution is the normal distribution, often represented by the bell curve. It is used to model everything from heights to test scores and is defined mathematically by a probability density function. Other examples include the exponential and uniform distributions. The continuous nature of these distributions means probabilities are calculated over intervals, rather than at specific points.

Expected Value

The expected value, also known as the mean, is the long-run average value of repetitions of the experiment it represents. It's a central concept in probability and statistics and serves as a measure of the center of the distribution of the random variable.

Mathematically, for a discrete random variable, the expected value is calculated by summing the product of each possible value the random variable can assume and the probability of that value. In contrast, for a continuous random variable, the expected value is calculated using an integral over the entire range of the variable. It's crucial to remember that the expected value might not correspond to any of the actual outcomes and may not even be a possible value for the random variable.

Variance

The concept of variance quantifies the spread of a distribution; it's a numerical value that describes how much the values of a random variable differ from the expected value (or mean). A low variance indicates that the data points are generally close to the mean, while a high variance signifies that the data points are spread out over a wider range of values.

Mathematically, variance is the average of the squared differences from the expected value. The calculation involves squaring the deviations, which gives more weight to extreme differences. This is why outliers can significantly affect the variance. In most practical applications, the square root of variance, known as the standard deviation, is used to represent variability because it is in the same units as the data.