The square of a standard normal variate is a
Chi-square distribution
Understanding the Square of a Standard Normal Variate
A standard normal variate, often denoted by \(Z\), is a random variable that follows the standard normal distribution. This distribution has a mean of 0 and a variance of 1. In mathematical notation, we write \(Z \sim N(0, 1)\).
The question asks about the distribution of the square of such a variable, which is \(Z^2\).
Distribution of \(Z^2\)
In probability and statistics, there is a well-known result concerning the distribution of the square of a standard normal variate. This result is fundamental in the construction of several important statistical tests and distributions.
- If \(Z\) is a single random variable following the standard normal distribution \(N(0, 1)\), then the random variable \(Y = Z^2\) follows a chi-square distribution.
- The degrees of freedom for the chi-square distribution are determined by the number of independent standard normal variates that are squared and summed. Since we are only squaring a single standard normal variate \(Z\), the resulting chi-square distribution has 1 degree of freedom.
Thus, the distribution of the square of a standard normal variate is a chi-square distribution with 1 degree of freedom, denoted as \(\chi^2(1)\).
Evaluating the Options
Let's briefly look at the other options provided to understand why they are not the correct distribution for the square of a standard normal variate:
- Beta distribution: This distribution is typically used to model probabilities or proportions, defined on the interval [0, 1]. The square of a standard normal variate \(Z^2\) can take any non-negative value (\([0, \infty)\)), not just values between 0 and 1.
- Cauchy distribution: The Cauchy distribution is known for having heavy tails and undefined moments (like mean and variance). The square of a standard normal variate has a well-defined mean (equal to its degrees of freedom, which is 1) and variance (equal to 2 times its degrees of freedom, which is 2).
- F-distribution: An F-distribution arises from the ratio of two independent chi-square variables, each divided by their respective degrees of freedom. It is used in ANOVA and other tests comparing variances, but it is not the distribution of a single squared standard normal variate.
Based on the definition and properties, the square of a standard normal variate is indeed a chi-square distribution with 1 degree of freedom.
Important Questions from Random Variables Basics
Let X1 and X2 be two independent exponentially distributed random variables with means 0.5 and 0.25, respectively. Then Y = min (X1, X2) is
An automobile plant contracted to buy shock absorbers from two suppliers X and Y. X supplies 60% and Y supplies 40% of the shod absorbers. All shock absorbers are subjected to a quality test. The ones that pass the quality test are considered reliable Of X's shock absorbers, 96% are reliable. Of Y's shock absorbers, 72% are reliable. The probability that a randomly chosen shock absorber, which is found to be reliable is made by Y is
For the function f(x) = a + bx, 0 ≤ x ≤ 1, to be a valid probability density function, which one of the following statements is correct?
If f(𝑥) and g(𝑥) are two probability density functions,
\(\begin{array}{l} f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{x}{a} + 1}&{: - a \le x < 0}\\ { - \frac{x}{a} + 1}&{0 \le x \le a}\\ 0&{otherwise} \end{array}} \right.\\ g\left( x \right) = \left\{ {\begin{array}{*{20}{c}} { - \frac{x}{a}}&{:-a \le x \le 0}\\ {\frac{x}{a}}&{:0 \le x \le a}\\ 0&{:otherewise} \end{array}} \right. \end{array}\)
Which one of the following statements is true?
Let the probability density function of a random variable x be given as
f(x) = ae-2|x|
The value of ‘a’ is __________.
