Probability Distributions

Test your knowledge of probability distributions with this comprehensive Probability Distribution Quiz! Covering exponential, normal, gamma, binomial, and chi-square distributions, this quiz is perfect for students, researchers, statisticians, and data analysts preparing for exams or interviews. Practice key concepts like mean, variance, Z-scores, kurtosis, and distribution properties to strengthen your statistical skills. Let us start with the online Probability Distribution Quiz now.

Online Probability Distribution Quiz with Answers

• If the mean of the exponential distribution is 2, then its variance is
• If the mean of the exponential distribution is 2, then the sum of 10 such independent variates will follow a gamma distribution with mean
• If the mean of the exponential distribution is 2, then the sum of 10 such independent variates will follow a gamma distribution with variance
• Which of the following is not a characteristic of a normal distribution?
• The total area under a normal distribution curve to the left of the mean is always
• The tails of the normal distribution
• For a normal distribution, the mean is 40 and the standard deviation is 8. The value of $Z$ for $X=52$ is
• What is the area under a conditional Cumulative Density Function?
• The mineral content of a particular brand of supplement pills is normally distributed, with a mean of 490 mg and a variance of 400. What is the probability that a randomly selected pill contains at least 500 mg of minerals?
• If the $Z$ (standard variable) score of a value is 1.5, it means the value is
• If the shape of the data is leptokurtic, then it must be
• A flat peak symmetrical curve is called —————-.
• Another name for the bell-shaped normal curve is ——————.
• A variable whose mean is zero and variance is one is called
• If the shape of the data is bell-shaped normal, which of the following statements must be true
• If a random variable $Y$ is distributed as normal with mean 0 and variance equal to 1, then $Y^2$ will be distributed as
• If $X$ and $Y$ are two independently distributed standard normal variables, then $X^2+Y^2$ will be distributed as ————–.
• If $X$ and $Y$ are two independently distributed standard normal variables, then $\frac{X^2}{Y^2}$ will be distributed as —————-.
• The sum of squares of a sequence of independent normal variates with mean $\mu$ and variance $\sigma^2$ is said to be
• In a binomial distribution, if $p$, $q$, and $n$ are the  probability of success, failure, and number of trials, respectively, then the variance is given by

R Frequently Asked Questions

In this post, I will discuss some common shape of data distributions. Data distributions can take on a variety of shapes, which can provide insights into the underlying characteristics of the data. By examining the shape of data distributions, professionals can gain insights that guide decision-making, improve processes, and enhance predictive accuracy in various fields.

Normal Distribution

A normal distribution of data possesses the following characteristics:

• Symmetrical and bell-shaped.
• Mean, median, and mode are all equal in a symmetric/normal distribution.
• Approximately 68% of the data falls within one standard deviation from the mean.

Symmetric – The data distribution is approximately the same shape on either side of a central dividing line.

Examples of normal distributions are: Men’s Heights and SAT Math scores.

Skewed Distribution

• Right (Positive) Skew: The tail on the right side is longer or fatter. Mean > median. In other words, a few data values are much higher than the majority of values in the set.  (Tail extends to the right). In right-skewed distributions, generally, Generally, the mean is greater than the median (and mode) in a right-skewed distribution. Personal Income in Pakistan and Men’s weight are examples of right positive skewed distribution.
• Left (Negative) Skew: The tail on the left side is longer or fatter. Mean < median. In other words, A few data values are much lower than the majority of values in the set.  (Tail extends to the left). In left-skewed distributions, generally, the mean is less than the median (and mode) in a left-skewed distribution.

Uniform Distribution

In the uniform distribution, all data values are equally represented. In uniform distribution, every outcome is equally likely and the shape of uniform distribution is of Rectangular shape.

Bimodal Distribution

A bimodal distribution has two distinct peaks or modes. It indicates the presence of two different sub-populations within the data.

Multimodal Distribution

Multimodal distributions are similar to bimodal but with more than two peaks. This distribution suggests even more complex underlying groupings.

Exponential Distribution

Exponential distributions often represent the time until an event occurs (e.g., waiting times) and are characterized by a rapid decline in probability.

Binomial Distribution

The binomial distribution represents the number of successes in a fixed number of trials. It is a discrete distribution with only two mutually exclusive and collectively exhaustive outcomes (success/failure).

Poisson Distribution

The Poisson distribution represents the number of events occurring within a fixed interval of time or space. It is useful for counting occurrences of rare events.

Note that Each shape has its implications for statistical analysis and helps in selecting appropriate techniques for data analysis. Understanding these distributions is crucial for interpreting data accurately.

Key Applications of Shape of Data Distributions

Some of the key applications of Shape of Data Distributions are:

1. Statistical Analysis
   • The shape of Data Distributions helps in selecting appropriate statistical tests (parametric vs. non-parametric) based on the normality of data.
   • Normal distributions allow for the use of techniques like t-tests, z-tests, and ANOVA.
2. Risk Management
   • In finance, the return distributions of assets are analyzed to assess risks and make informed investment decisions.
   • Non-normal distributions can indicate higher risks, impacting portfolio management.
3. Quality Control
   • In manufacturing, control charts are used to monitor processes; the distribution shape indicates whether a process is stable or in control.
   • Detects defects and variations in production processes.
4. Epidemiology
   • Distribution shapes can model the spread of diseases, helping to predict outbreaks and understand transmission patterns.
   • Bimodal or multimodal distributions can indicate multiple populations affected differently.
5. Machine Learning
   • Many algorithms assume a certain distribution of the data (e.g., Gaussian distribution).
   • Understanding the distribution shape can help in feature selection and engineering.
6. Psychometrics and Social Sciences
   • Assessing test scores or survey responses can reveal insights into populations (e.g., identifying bias).
   • Skewed distributions can indicate social inequality or access issues.
7. Environmental Studies
   • Used to assess environmental data, like rainfall patterns or pollution levels, which often do not follow a normal distribution.
   • Helps in formulating regulations and responses based on the observed distribution.
8. Marketing and Customer Behavior
   • Analyzing purchase distributions to understand customer preferences and segmentation.
   • Helps in tailoring marketing strategies based on consumer behavior patterns.

Online Quiz Website with Answers

The post is about the MCQs Probability Distributions Quiz. There are 20 multiple-choice questions about probability distributions covering distributions such as discrete and continuous Binomial Probability Distribution, Bernoulli Probability Distribution, Poisson Probability Distribution, Poisson Probability, Distribution, Geometric Probability Distribution, Hypergeometric Probability Distribution, Chi-Square distribution, Normal distribution, and F-distribution. Let us start with the MCQs Discrete Probability Distributions Quiz.

Please go to Probability Distribution Quiz 8 to view the test

Online Probability Distribution Quiz

• You find a z-score of -1.99. Which statement(s) is/are true?
• Expected values are properties of what?
• If you got a 75 on a test in a class with a mean score of 85 and a standard deviation of 5, the z-score of your test score would be
• The spread of the normal curve depends upon the value of:
• Which of the following can best be described as a normal distribution?
• In its standardized form, the normal distribution
• A test is administered annually. The test has a mean score of 150 and a standard deviation 20. If Chioma’s z-score is 1.50, what was her score on the test?
• The P-value for a normally distributed right-tailed test is P=0.042. Which of the following is INCORRECT?
• The time X taken by a cashier in a grocery store express lane to complete a transaction follows a normal distribution with a mean of 90 seconds and a standard deviation of 20 seconds. What is the first quartile of the distribution of X (in seconds)?
• Green sea turtles have normally distributed weights, measured in kilograms, with a mean of 134.5 and a variance of 49.0. A particular green sea turtle’s weight has a z-score of -2.4. What is the weight of this green sea turtle? Round to the nearest whole number.  
• We look for a model, as realistic as possible, for a continuous random variable $X$ that represents the lifetime of a machine, and whose mean and variance are equal to 1 and 3, respectively. Which of the following distributions can be acceptable?
  Uniform
  Exponential
  Gamma
  Gaussian
• The square of a Gaussian N(1, 3)
• The distribution function of the random variable $X$ is given by $F_X(x)=1-\frac{1}{x^2}$ for $x \ge c$, 0 otherwise, where $c$ is a constant. What is the set of possible values of the constant $c$?
• A random variable $Y$ has the following distribution y:     -1   0   1    2 p(y):  3C 2C 0.4 0.1 The value of the constant C is
• If $Z$ has a standard normal distribution, if $U$ has a chi-square distribution with $k$ degrees of freedom and if $Z$ and $U$ are independent then the distribution of $X=\frac{Z}{\sqrt{\frac{U}{\sqrt{k}}}}$ is
• If $X$ is a F-distributed random variable with $m$ and $n$ df, then $W=\frac{mX/n}{1+mX/n}$ has a
• The number of parameters in multivariate normal distribution having $p$ variables are
• The moment generating function of Gamma distribution with parameter $\lambda$ and $k$ is
• The moment generating function of normal distribution is
• When the experiment is repeated a variable number of times to obtain a fixed number of successes is
• If the mean of the Chi-Square distribution is 4 then its variance is

MCQs General Knowledge