Probability Distributions

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

Shape of Data Distributions

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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.

Shape of Data Distributions

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

Probability Distribution Quiz 8

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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.

MCQs Probability Distribution Quiz

Probability Distribution Quiz with Answers

1. 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?

 
 
 
 

2. 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 $c$?

 
 
 
 

3. The P-value for a normally distributed right-tailed test is P=0.042. Which of the following is INCORRECT?

 
 
 
 

4. The number of parameters in multivariate normal distribution having $p$ variables are

 
 
 
 

5. 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

 
 
 
 

6. 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)
 
 
 
 

7. If the mean of the Chi-Square distribution is 4 then its variance is

 
 
 
 

8. 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)?

 
 
 
 

9. The moment generating function of Gamma distribution with parameter $\lambda$ and $k$ is

 
 
 
 

10. Which of the following can best be described as a normal distribution?

 
 
 
 

11. Expected values are properties of what?

 
 
 
 

12. 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

 
 
 
 

13. 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.

 
 
 
 

14. In its standardized form, the normal distribution

 
 
 
 

15. If $X$ is a F-distributed random variable with $m$ and $n$ df, then $W=\frac{mX/n}{1+mX/n}$ has a

 
 
 
 

16. When the experiment is repeated a variable number of times to obtain a fixed number of success is

 
 
 
 

17. You find a z-score of -1.99. Which statement(s) is/are true?

 
 
 
 

18. The moment generating function of normal distribution is

 
 
 
 

19. The spread of the normal curve depends upon the value of:

 
 
 
 

20. 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

 
 
 
 

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

Solved Binomial Distribution Questions

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This post is about some solved Binomial distribution Questions. These solved binomial distribution questions make use of computation of (i) the exact probability case, (ii) at least case, (iii) at most case, and (iv) between cases.

Binomial-Probability-Distribution Binomial distribution questions
  • The sum of all probabilities in the distribution sums up to 1
  • The probability of success in all $n$ trials is $p^n$
  • The probability of failure in all $n$ trials is $(1 – p)^n = q^n$
  • Probability of success in at least one trial = $P(X \ge 1) = 1 – P(X = 0) = 1 – q^n$.
  • Probability of at least $x$ successes = $P(X \ge x) = \sum\limits_{x} \binom{n}{x}p^xq^{n-x}\quad (x = x, x + 1,\cdots, n$)
  • Probability of at most $x$ successes = $P(X \le x) =\sum\limits_{x} \binom{n}{x}p^x q^{n-x}\quad (x=0,1,\cdots,x)$
  • If in $n$ trials, the experiment is repeated $N$ times, the expected frequencies are $N\cdot P(x)$ for $x = 0, 1, 2, 3, \cdots, n$.

Solved Binomial Distribution Questions

Question 1: A die is rolled 5 times and a 5 or 6 is considered a success. Find the probability of (i) no success, (ii) at least 2 successes, (iii) at least one but not more than 3 successes.

Solution:

The Sample Space is $S=\{1, 2, 3, 4, 5, 6\}$. Since the occurrence of 5 or 6 is considered a success, therefore, $p=\frac{2}{6}=\frac{1}{3} \Rightarrow q=1-p = 1-\frac{1}{3} = \frac{2}{3}$.

(i) No success

$n=5, p=\frac{1}{3}, q=\frac{2}{3}$, x=0$

\begin{align*}
P(X=x) &= \binom{n}{x}p^x q^{n-x}\\
P(X=0) &= \binom{5}{0} \left(\frac{1}{3}\right)^0\left(\frac{2}{3}\right)^5=0.1316
\end{align*}

(ii) At least 2 successes

\begin{align*}
P(X \ge 2) & = 1 – P(X<2)\\
&= 1 – [P(X=0) + P(X=1)]\\
&= 1- [0.13168 + 0.3292] = 0.5391
\end{align*}

(iii) At least one but not more than 3 successes

\begin{align*}
P(1 \le x \le 3) &= P(X=1) + P(X=2) + P(X=3)\\
&= 0.3292 + 0.3292 + 0.1646 = 0.823
\end{align*}

Question 2: Find the probability of getting (i) exactly 4 heads and (ii) not more than 4 heads when 6 coins are tossed.

Solution:

From the given information, $n = 6, x = 4, p = q = \frac{1}{2}$

(i) Exactly 4 heads

\begin{align*}
P(X=x) &= \binom{n}{x} p^x q^{n-x}\\
&= \binom{6}{4} \left(\frac{1}{2}\right)^4 \left(\frac{1}{2}\right)^{6-4} = 0.234
\end{align*}

(ii) Not more than 4 heads

\begin{align*}
P(X\le 4) & = 1 – p(X\ge 4)\\
&= 1 – P(X=4) + P(X=5) + P(X=6)
\end{align*}

Question 3: If 60% of the voters in a large district prefer candidate-A, what is the probability that in a sample of 12 voters, exactly 7 will prefer A?

Solution:

From given information in the questions, $p=06, q=0.4, n=12, x=7$

\begin{align*}
P(X=x)&= \binom{n}{x}p^x q^{n-x}\\
P(X=7) &= \binom{12}{7} (0.6)^7(0.4)^5&= 0.227
\end{align*}

Question 4: The probability that a patient recovers from a delicate heart operation is 0.9. What is the probability that exactly 5 of the next 7 patients having this operation survive?

Solution:

From the given information in the question, $n=7, x=5, p=0.9, q=0.10$

\begin{align*}
P(X=x)&= \binom{n}{x}p^x q^{n-x}\\
P(X=5) &= \binom{7}{5}(0.9)^5(0.1)^2 = 0.124
\end{align*}

Question 5: The incidence of occupational disease in an industry is such that the workmen have a 20% chance of suffering from it. What is the probability that out of 6 workmen (i) not more than 2, and (ii) 4 or more will catch the disease?

Solution:

From the given information in the questions

Probability of suffering from occupational disease = $\frac{20}{100}=\frac{1}{5}=0.20$

Probability of not suffering from occupational disease = $1 – \frac{1}{5} = \frac{4}{5}=0.80$

(i) Probability that out of 6 workers, not more than two will suffer

\begin{align*}
P(X\le 2) &= \binom{6}{0}\left(\frac{4}{5}\right)^0 \left(\frac{1}{5}\right)^6 + \binom{6}{1}\left(\frac{4}{5}\right)^1 \left(\frac{1}{5}\right)^5 + \binom{6}{2}\left(\frac{4}{5}\right)^2 \left(\frac{1}{5}\right)^4\\
&=0.01696
\end{align*}

(ii) Probability that out of 6 workers, 4 or more will suffer

\begin{align*}
P(X\ge 4) &= \binom{6}{4}\left(\frac{4}{5}\right)^4 \left(\frac{1}{5}\right)^2 + \binom{6}{5}\left(\frac{4}{5}\right)^5 \left(\frac{1}{5}\right)^1 + \binom{6}{6}\left(\frac{4}{5}\right)^6 \left(\frac{1}{5}\right)^0\\
&=0.90112
\end{align*}

Question 6: A multiple-choice has 15 questions, each with 4 possible answers of which only 1 is the correct answer. What is the probability that sheer guesswork yields from 5 to 10 correct answers?

Solution:

Probability of answering any question correctly: $p=\frac{1}{4}=0.25$

Probability of answering any question wrongly: $q=\frac{3}{4}=0.75$

\begin{align*}
P(5 \le x \le 10) &= P(X=5) + P(X=6) + \cdots + P(X=10)\\
&=\binom{15}{5}\left(\frac{1}{4}\right)^5\left(\frac{3}{4}\right)^{10}+\cdots + \binom{15}{10}\left(\frac{1}{4}\right)^5\left(\frac{3}{4}\right)^{5} \\
&= 0.31339
\end{align*}

Question 7: A commuter drivers to work each morning. The route she takes each day includes ten stoplights. Assume the probability each stoplight is red when she gets to it is 0.2 and that these stoplights (trials) are independent. What is the distribution of $X$, the number of times she must stop for a red light on her way to work? Evaluates $P(X=0) and $P(X<3).

Solution:

The distribution of $X$ is binomial because trials are independent. The probability of getting red spotlight (success) is 0.2 which remains the same, the number of trials is fixed ($n=10$).

The further information given in the Question is: $n=10, p=0.2, q=0.8$

\begin{align*}
P(X=0)&=\binom{10}{0} (0.2)^0(0.8)^{10} = 0.10737\\
P(X<3) &=\binom{10}{0} (0.2)^0(0.8)^{10} + \binom{10}{1} (0.2)^1(0.8)^{9} + \binom{10}{2} (0.2)^2(0.8)^{8} = 0.6777
\end{align*}

Application of Binomial Probability Distribution

  • Quality Control:
    • Assessing Product Reliability: Manufacturers use binomial distribution to estimate the probability of defective products in a batch, helping them maintain quality standards.
    • Predicting Failure Rates: By analyzing past data, companies predict the likelihood of equipment failure using a binomial probability distribution, aiding in preventive maintenance and reducing downtime.
  • Genetics:
    • Predicting Inheritance Patterns: In genetics, Binomial distribution helps to predict the probability of offspring inheriting specific traits based on parental genotypes.
    • Analyzing Genetic Mutations: Binomial distribution is used to study the frequency of genetic mutations in populations.
  • Medicine:
    • Clinical Trials: Binomial distribution is essential for designing and analyzing clinical trials, assessing the effectiveness of treatments, and determining the probability of side effects.
    • Epidemiology: Binomial distribution helps to model the spread of infectious diseases and predict outbreak risks.
  • Finance:
    • Risk Assessment: Financial institutions use Binomial Probability Distribution to assess the risk of loan defaults or investment failures.
    • Option Pricing: Binomial probability distribution is a key component of option pricing models, helping to determine the fair value of options contracts.
  • Social Sciences:
    • Survey Analysis: Binomial distribution is used to analyze survey data, such as predicting voter behavior or public opinion on specific issues.
    • Market Research: Binomial Probability Distribution helps businesses to understand consumer preferences and predict market trends.

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