Probability Distributions

Introduction to Probability Distributions, Discrete Probability, Continuous Probability, Distribution Functions, Density Functions, Real life Examples of Probability Distributions

Normal Probability Distribution

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The Gaussian or normal probability distribution role is very important in statistics. It was investigated by researchers/ persons interested in gambling or in the distribution of errors made by people observing astronomical events. The normal probability distribution is important in other fields such as social sciences, behavioural statistics, business and management sciences, and engineering and technologies.

Importance of Normal Distribution

Some of the important reasons for the normal probability distribution are:

  • Many variables such as (weight, height, marks, measurement errors, IQ, etc.) are distributed as the symmetrical bell-shaped normal curve approximately.
  • Many inferential procedures (parametric tests: confidence intervals, hypothesis testing, regression analysis, etc.) assume that the variables follow the normal distribution.
  • All probability distributions approach a normal distribution under some conditions.
  • Even if a variable is not normally distributed, a distribution of sample sums or averages on that variable will be approximately normally distributed if the sample size is large enough.
  • The mathematics of a normal curve is well-known and relatively simple. One can find the probability that a score randomly sampled from a normal distribution falls within the interval $a$ and $b$ by integrating the normal probability density function (PDF) from $a$ to $b$. This is equivalent to finding the area under the curve between $a$ and $b$ assuming a total area of one.
  • Due to the Central Limit Theorem, the average of many independent random variables tends to follow a normal probability distribution, regardless of the original distribution of the variables.

Probability Density Functions of Normal Distribution

The probability density function known as the normal curve. $F(X)$ is the probability density, aka the height of the curve at value $X$.

$$F(X) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(X-\mu)^2}{2\sigma^2} }$$

There are two parameters in the PDF of normal distribution, (i) the mean and (ii) the standard deviation. Everything else in the PDF of normal distribution on the right-hand side is a constant. There is a family of normal probability distribution with respect to their means and their standard deviations.

Standard Normal Probability Distribution

One can work with normal curve, even if one don’t know about integral calculus. One can use computer to compute the area under the normal curve or make use of the normal curve table. The normal curve table (standard normal table) is based on the standard normal curve ($Z$), which has a mean of 0 and a variance of 1. To use a standard normal curve table, one need to convert the raw score to $Z$-scores. A $Z$-score is the number of standard deviations ($\sigma$ or $s$) a score is above or below the mean of a reference distribution.

$$Z_X = \frac{X-\mu}{\sigma}$$

For example, suppose one wish to know the percentile rank of a score of 90 on an IQ test with $\mu = 100$ and $\sigma=10$. The $Z$-score will be

$$Z=\frac{X-\mu}{\sigma} = \frac{90-100}{10} = -1$$

One can either integrate the normal cure from $-\infty$ to $-1$ or use the standard normal table. The probability or area under the curve on the left of $-1$ is 0.1587 or 15.87%.

Standard Normal Probability distribution Curve

Key Characteristics of Normal Probability Distribution

  • Symmetry: In normal probability distribution, the mean, median, and mode are all equal and located at the center of the curve.
  • Spread: In normal distribution, the spread of the data is determined by the standard deviation. A larger standard deviation means that the curve is wider, and a smaller standard deviation means a narrower curve.
  • Area under the Normal Curve: The total area under the normal curve is always equal to 1 or 100%.
normal curve dnorm() normal probability distribuiton

Real-Life Applications of Normal Distribution

The following are some real-life applications of normal probability distribution.

  • Natural Phenomena:
    • Biological Traits: Many biological traits, such as weight, height, and IQ scores, tend to follow a normal distribution. This helps us to understand the typical range of values for different biological traits and identify outliers.
    • Physical Measurements: Errors in measurements often follow a normal distribution. This knowledge is crucial in fields like engineering and physics for quality control and precision.
  • Statistical Inference:
    • Hypothesis Testing: The normal distribution is used extensively in hypothesis testing to determine the statistical significance of the results. By understanding the distribution of sample means, one can make inferences about population parameters.
    • Confidence Intervals: Normal distribution helps calculate confidence intervals, which provide a range of values within which a population parameter is likely to fall with a certain level of confidence.
  • Machine Learning and Artificial Intelligence:
    • Feature Distribution: Many machine learning (ML) algorithms assume that features in data follow a normal distribution. The normality assumption about machine learning algorithms can influence the choice of algorithms and the effectiveness of models.
    • Error Analysis: The normal distribution is used to analyze the distribution of errors in machine learning models, helping to identify potential biases and improve accuracy.
  • Finance and Economics:
    • Asset Returns: While not perfectly normal, many financial assets, such as stock prices, follow an approximately normal distribution over short time periods. The assumption of normality is used in various financial models and risk assessments.
    • Economic Indicators: Economic indicators such as GDP growth rates and inflation rates often exhibit a normal distribution, allowing economists to analyze trends and make predictions.
  • Quality Control:
    • Process Control Charts: In manufacturing and other industries, normal distribution is used to create control charts that monitor the quality of products or processes. By tracking the distribution of measurements, one can identify when a process is going out of control.
  • Product Quality: Manufacturers use statistical quality control methods based on normal distribution to ensure that products meet quality standards.
  • Everyday Life:
    • Standardized Tests: The standardized Test scores, such as SAT and GRE, are often normalized to a standard normal distribution, allowing for comparisons between different test-takers.
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    Probability Distributions MCQS 6

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    The post is about Probability distributions MCQs with Answers. There are 20 multiple-choice questions covering topics related to binomial, Poisson, exponential, normal, gamma, standard normal, hypergeometric, and bivariable distributions. Let us start with the quiz probability distributions MCQs with answers.

    Online Multiple Choice Questions about Probability Distributions

    1. The moment generating function of Binomial distribution is

     
     
     
     

    2. If $Mean = Variance$ the distribution is called

     
     
     
     

    3. The continuous Random variable $X$ has a gamma distribution with Parameters $\alpha$ and $\beta$. The special gamma distribution for which $\alpha = 1$ is called.

     
     
     
     

    4. The continuous random variable $X$ has a gamma distribution with parameters $\alpha$ and $\beta$. The special gamma distribution for which $\alpha=\frac{v}{2}$ and $\beta=2$ where $v$ is a +ve integer is called.

     
     
     
     

    5. The number of products manufactured in a factory in a day is 3500 and the probability that some pieces are defective is 0.55 then the mean of the binomial probability distribution is

     
     
     
     

    6. A bivariate normal distribution has a number of parameters in it

     
     
     
     

    7. For a normal distribution, the measure of kurtosis equals to

     
     
     
     

    8. The area under the normal curve with $\mu\pm 2\sigma$ is

     
     
     
     

    9. In binomial distribution, the formula for calculating standard deviation is

     
     
     
     

    10. Which of the following is NOT an assumption of the Binomial distribution?

     
     
     
     

    11. If the value of $p$ is smaller or lesser than 0.5 then binomial distribution is classified as

     
     
     
     

    12. The maximum ordinate of the normal curve is at

     
     
     
     

    13. Standard normal probability distribution has a mean equal to 40, whereas the value of random variable x is 80 and the z-statistic is equal to 1.8, the standard deviation of the standard normal probability distribution is

     
     
     
     

    14. If $Mean > Variance$ then the distribution is

     
     
     
     

    15. The normal distribution will be less spread out when

     
     
     
     

    16. Consider probability distribution as standard normal, if the value of $\mu$ is 75, the value of $x$ is 120 with an unknown standard deviation of distribution then the value of z-statistic

     
     
     
     

    17. If the value of $x$ is less than $\mu$ of standard normal probability distribution then the

     
     
     
     

    18. If in a Gamma density, $k=1$ the Gamma density becomes

     
     
     
     

    19. A bank received 2600 applications for a home mortgage. The probability of approval is 0.78 then the standard deviation of the binomial probability distribution is

     
     
     
     

    20. Let $X$ be a positive random variable and let a new random variable $Y$ be defined as $Y=log X$. If $Y$ has a normal distribution then $X$ is

     
     
     
     

    Probability Distributions MCQS with Answers

    Probability Distributions MCQs Quiz with Answers
    • The number of products manufactured in a factory in a day is 3500 and the probability that some pieces are defective is 0.55 then the mean of the binomial probability distribution is
    • A bank received 2600 applications for a home mortgage. The probability of approval is 0.78 then the standard deviation of the binomial probability distribution is
    • Consider probability distribution as standard normal, if the value of $\mu$ is 75, the value of $x$ is 120 with an unknown standard deviation of distribution then the value of z-statistic
    • If the value of $x$ is less than $\mu$ of standard normal probability distribution then the
    • Standard normal probability distribution has a mean equal to 40, whereas the value of random variable x is 80 and the z-statistic is equal to 1.8, the standard deviation of the standard normal probability distribution is
    • In binomial distribution, the formula for calculating standard deviation is
    • If the value of $p$ is smaller or lesser than 0.5 then binomial distribution is classified as
    • The continuous Random variable $X$ has a gamma distribution with Parameters $\alpha$ and $\beta$. The special gamma distribution for which $\alpha = 1$ is called.
    • The continuous random variable $X$ has a gamma distribution with parameters $\alpha$ and $\beta$. The special gamma distribution for which $\alpha=\frac{v}{2}$ and $\beta=2$ where $v$ is a +ve integer is called.
    • For a normal distribution, the measure of kurtosis equals to
    • A bivariate normal distribution has a number of parameters in it
    • The moment generating function of Binomial distribution is
    • If $Mean > Variance$ then the distribution is
    • If $Mean = Variance$ the distribution is called
    • The area under the normal curve with $\mu\pm 2\sigma$ is
    • The maximum ordinate of the normal curve is at
    • Which of the following is NOT an assumption of the Binomial distribution?
    • The normal distribution will be less spread out when
    • Let $X$ be a positive random variable and let a new random variable $Y$ be defined as $Y=log X$. If $Y$ has a normal distribution then $X$ is
    • If in a Gamma density, $k=1$ the Gamma density becomes
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    Important MCQs Probability Distributions 4

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    The post is about MCQs Probability Distributions. There are 20 multiple-choice questions covering the topics related to Chi-Square distribution, F-distribution, Binomial distribution, Student’s t distribution, and properties of distributions. Let us start with MCQs Probability Distributions.

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    MCQs Probability Distributions Quiz

    MCQs Probability Distributions with Answers
    • A family of parametric distribution in which mean always greater than its variance is:
    • The family of parametric distributions which has a mean always less than variance is:
    • The family of parametric distributions for which moment generating function does not exist is:
    • The distribution for which the mode does not exist is:
    • The relation between the mean and variance of $\chi^2$ with $n$ degrees of freedom is
    • The $F$-distribution curve in respect of tails is:
    • If $X$ has a binomial distribution with parameter $p$ and $n$ then $\frac{X}{n}$ has the variance:
    • The binomial distribution is symmetrical if $p=p=?$
    • The shape of geometric distribution is
    • If $X\sim N(\mu, \sigma^2)$ and $a$ and $b$ are real numbers, then mean of $(aX+b)$ is
    • The distribution of sample correlation is
    • Events having an equal chance of occurrence are called
    • Student’s $t$-distribution curve is symmetrical about mean, it means that
    • The distribution possessing the memoryless property is
    • The chi-square distribution is used for the test of
    • The probability of failure in binomial distribution is denoted by
    • In binomial distribution, the formula for calculating the mean is  
    • In binomial probability distribution, dependents of standard deviations must include
    • The formula to calculate standardized normal random variables is
    • The mean of a binomial probability distribution is 857.6 and the probability is 64% then the number of values of binomial distribution
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    Best Probability Distribution Questions 3

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    The post is about Probability Distribution Questions. There are 20 multiple-choice questions covering topics related to normal probability distribution, standard normal probability distribution, and its properties. Let us start with the Quiz Probability Distribution Questions.

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    Probability Distribution Questions with Answers

    MCQs Probability Distribution Questions with Answers
    • The area under the normal curve on either side of the mean is
    • In the case of a symmetrical distribution
    • The mean deviation of Normal Distribution is
    • The Normal Distribution has parameters
    • In Normal distribution, the parameters which control the flatness of the curve is
    • We use normal distribution when $n$ is
    • The median of the normal distribution corresponds to the value of $Z$ equal to
    • The lower and upper quartiles of standard normal variate are respectively
    • The shape of the normal curve can be related to
    • The total Area under the normal curve is
    • Which of the following parameters controls the relative flatness of a normal distribution
    • In a normal distribution $E(X−\mu)^2$ is
    • If $X\sim N(55,49)$ then $\sigma$
    • The Normal Curve is asymptotic to the
    • The shape of the normal curve depends upon
    • If $X\sim N(16, 49)$, then mean is
    • Normal Distribution is
    • If $Y=5X + 10$ and $X$ is $N(10,25)$, then mean of $Y$ is
    • Normal Distribution is
    • The formula in which binomial distribution approaches normal probability distribution with the help of normal variable is written as
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