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‎Probability andStatistics/Prob Dist Functions/normal_dist.html‎

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<title>Normal Distribution Function</title>
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<body>
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<h1>Normal Distribution Function</h1>
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<p>The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric around its mean. It is often used to model natural phenomena and is characterized by its bell-shaped curve.</p>
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<h2>Properties of Probability Distribution Functions</h2>
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<ul>
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<li>The PDF is non-negative for all values of the random variable.</li>
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<li>The total area under the PDF curve is equal to 1, which represents the total probability.</li>
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<li>The probability that a random variable falls within a specific interval can be calculated by integrating the PDF over that interval.</li>
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</ul>
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<h2>Example: Normal Distribution</h2>
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<ol>
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<li>The normal distribution is a continuous probability distribution that is symmetric around its mean. It is often used to model natural phenomena and is characterized by its bell-shaped curve.</li>
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<li>The normal distribution is defined by two parameters: the mean (μ) and the standard deviation (σ). The mean determines the center of the distribution, while the standard deviation controls the spread of the data.</li>
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</ol>
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<p>One of the most well-known probability distribution functions is the normal distribution. It is often used to model natural phenomena and is characterized by its bell-shaped curve. The normal distribution is defined by two parameters: the mean (μ) and the standard deviation (σ). The mean determines the center of the distribution, while the standard deviation controls the spread of the data.</p>
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<p>The normal distribution, also known as the Gaussian distribution, is a common example of a probability distribution function. It is characterized by its bell-shaped curve and is defined by its mean (μ) and standard deviation (σ). The PDF of a normal distribution is given by:</p>
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<pre>
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f(x) = (1 / (σ * sqrt(2π))) * exp(-0.5 * ((x - μ) / σ)^2)
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</pre>
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<p>In this formula, exp() represents the exponential function, and sqrt() represents the square root function. The normal distribution is widely used in statistics and various fields due to its properties and applicability to real-world data.</p>
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</body>
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</html>

‎Probability andStatistics/Prob Dist Functions/poisson_dist_function.html‎

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<title>Poisson Distribution Function</title>
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<body>
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<h1>Poisson Distribution Function</h1>
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<h2>Probability Distribution Function</h2>
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<p>A probability distribution function (PDF) is a mathematical function that describes the likelihood of a random variable taking on a particular value. It provides a way to model and analyze the behavior of random phenomena. The PDF is defined for continuous random variables and is used to calculate probabilities for specific intervals.</p>
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<p>In contrast, a probability mass function (PMF) is used for discrete random variables and assigns probabilities to specific values. The PMF is defined for discrete random variables and is used to calculate probabilities for specific values.</p>
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<p>The main examples of probability mass functions include the Poisson distribution, which is used for modeling the number of events in a fixed interval, and the binomial distribution, which is used for modeling the number of successes in a fixed number of trials.</p>
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<p>In this lecture, we will focus on the Poisson distribution, which is a discrete probability distribution that describes the number of events occurring within a fixed interval of time or space, given that the events occur with a known constant mean rate and independently of the time since the last event. The Poisson distribution is often used to model rare events, such as the number of phone calls received at a call center in an hour or the number of accidents occurring at a traffic intersection in a day.</p>
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<h2>Poisson Distribution Function</h2>
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<p>The Poisson distribution is a discrete probability distribution that describes the number of events occurring within a fixed interval of time or space, given that the events occur with a known constant mean rate and independently of the time since the last event. The Poisson distribution is often used to model rare events, such as the number of phone calls received at a call center in an hour or the number of accidents occurring at a traffic intersection in a day.</p>
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<pre>
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P(X = k) = (λ^k * e^(-λ)) / k!
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</pre>
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<p>In this formula, k represents the number of events, λ represents the average rate of occurrence, e is the base of the natural logarithm, and k! is the factorial of k. The Poisson distribution is widely used in various fields, including statistics, engineering, and natural sciences, to model and analyze count data.</p>
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<h2>Properties of Poisson Distribution</h2>
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<ul>
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<li>The Poisson distribution is defined for non-negative integer values (0, 1, 2, ...).</li>
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<li>The mean and variance of a Poisson distribution are both equal to λ, where λ is the average rate of occurrence.</li>
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<li>The probability mass function (PMF) of a Poisson distribution is given by:</li>
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</ul>
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<pre>
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<ol>
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<li>Example1: If a call center receives an average of 5 calls per hour, the probability of receiving exactly 3 calls in an hour can be calculated using the Poisson distribution formula with λ = 5 and k = 3.
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P(X = 3) = (5^3 * e^(-5)) / 3! = (125 * e^(-5)) / 6 ≈ 0.1404</li>
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<li>Example2: If a traffic intersection experiences an average of 2 accidents per day, the probability of having no accidents in a day can be calculated using the Poisson distribution formula with λ = 2 and k = 0.
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P(X = 0) = (2^0 * e^(-2)) / 0! = (1 * e^(-2)) / 1 ≈ 0.1353</li>
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<li>Example3: If a website receives an average of 10 hits per minute, the probability of receiving more than 15 hits in a minute can be calculated using the Poisson distribution formula with λ = 10 and k > 15. This can be calculated as:
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P(X > 15) = 1 - P(X ≤ 15) = 1 - (P(X = 0) + P(X = 1) + ... + P(X = 15))</li>
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</ol>
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</pre>
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<h2> Mean and Variance</h2>
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<p>In a Poisson distribution, the mean (λ) and variance are equal. This means that if the average rate of occurrence is λ, then the expected number of events in a given interval is also λ, and the variability around that mean is also λ. This property makes the Poisson distribution unique among probability distributions and allows for straightforward calculations of probabilities and expectations.</p>
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<p>For example, if a call center receives an average of 5 calls per hour, the mean and variance of the number of calls received in an hour would both be 5. This means that on average, we can expect to receive 5 calls per hour, and the variability around that average is also 5.</p>
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<h2>First Moment (Mean)</h2>
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<ol>
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<li>The first moment of a probability distribution is the expected value or mean of the distribution. It provides a measure of the central tendency of the distribution.</li>
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<li>In the case of a Poisson distribution, the first moment is equal to its parameter λ, which represents the average rate of occurrence.</li>
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</ol>
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<h2>Second Moment (Variance)</h2>
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<ol>
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<li>The second moment of a probability distribution is the variance, which measures the spread or variability of the distribution around its mean.</li>
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<li>For a Poisson distribution, the second moment (variance) is also equal to its parameter λ. This means that the variability of the number of events in a given interval is directly related to the average rate of occurrence.</li>
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</ol>
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</body>
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‎Probability andStatistics/Prob Dist Functions/prob_mass_function.html‎

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<title>Probability Mass Function</title>
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<body>
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<h1>Probability Mass Function</h1>
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<p>A probability mass function (PMF) is a mathematical function that describes the probability distribution of a discrete random variable. It assigns probabilities to each possible value of the random variable, indicating the likelihood of each outcome occurring. The PMF is defined for discrete random variables and is used to calculate probabilities for specific values.</p>
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<h2>Properties of Probability Mass Functions</h2>
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<ul>
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<li>The PMF is non-negative for all values of the random variable.</li>
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<li>The sum of the probabilities assigned by the PMF over all possible values of the random variable is equal to 1, which represents the total probability.</li>
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<li>The probability that a random variable takes on a specific value can be directly obtained from the PMF.</li>
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</ul>
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<h2>Example: Binomial Distribution</h2>
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<p>One common example of a probability mass function is the binomial distribution. It models the number of successes in a fixed number of independent Bernoulli trials, where each trial has two possible outcomes: success or failure. The PMF of a binomial distribution is given by:</p>
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<pre>
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P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
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OR
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P(X = k) = (n! / (k! * (n - k)!)) * p^k * (1 - p)^(n - k)
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Here, (n choose k) is the binomial coefficient, which can be calculated as n! / (k! * (n - k)!), where n is the number of trials and k is the number of successes. The parameter p represents the probability of success on each trial. The binomial distribution is widely used in statistics and various fields to model scenarios involving binary outcomes.</pre>
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<ul>
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<li>Let's break down the formula for the PMF of a binomial distribution:</li>
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<li>n: The total number of independent trials.</li>
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<li>k: The number of successes we want to calculate the probability for.</li>
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<li>p: The probability of success on each trial.</li>
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<li>(n choose k): The binomial coefficient, which represents the number of ways to choose k successes from n trials. It can be calculated using the formula n! / (k! * (n - k)!).</li>
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<li>p^k: The probability of having k successes.</li>
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<li>(1 - p)^(n - k): The probability of having (n - k) failures, which is the complement of success.</li>
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</ul>
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</pre>
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<pre>
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<ol>
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<li>Example1: If we flip a fair coin 10 times, the probability of getting exactly 3 heads can be calculated using the binomial distribution formula with n = 10, k = 3, and p = 0.5.
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P(X = 3) = (10 choose 3) * (0.5)^3 * (1 - 0.5)^(10 - 3)
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= (10! / (3! * (10 - 3)!)) * (0.5)^3 * (0.5)^7
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= (120) * (0.125) * (0.0078125)
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≈ 0.1172</li>
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<li>Example2: If we flip a fair coin 10 times, the probability of getting exactly 5 heads can be calculated using the binomial distribution formula with n = 10, k = 5, and p = 0.5.
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P(X = 5) = (10 choose 5) * (0.5)^5 * (1 - 0.5)^(10 - 5)
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= (10! / (5! * (10 - 5)!)) * (0.5)^5 * (0.5)^5
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= (252) * (0.03125) * (0.03125)
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≈ 0.2461</li>
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<li>Example3: If we flip a fair coin 10 times, the probability of getting at least 7 heads can be calculated using the binomial distribution formula with n = 10, k ≥ 7, and p = 0.5. This can be calculated as:
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P(X ≥ 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)</li>
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<li>Example4: If we flip a fair coin 10 times, the probability of getting at most 2 heads can be calculated using the binomial distribution formula with n = 10, k ≤ 2, and p = 0.5. This can be calculated as:
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P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)</li>
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<li>Example5: If we flip a fair coin 10 times, the probability of getting between 3 and 6 heads (inclusive) can be calculated using the binomial distribution formula with n = 10, 3 ≤ k ≤ 6, and p = 0.5. This can be calculated as:
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P(3 ≤ X ≤ 6) = P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)</li>
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</ol>
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</pre>
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<p>In this formula, n represents the number of trials, k represents the number of successes, p represents the probability of success on an individual trial, and (n choose k) is the binomial coefficient. The binomial distribution is widely used in statistics and various fields to model scenarios involving binary outcomes.</p>
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<h2>First Moment (Mean)</h2>
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<ol>
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<li>The first moment of a probability distribution is the expected value or mean of the distribution. It provides a measure of the central tendency of the distribution.</li>
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<li>In the case of a binomial distribution, the first moment (mean) is given by n * p, where n is the number of trials and p is the probability of success on each trial.</li>
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</ol>
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<h2>Second Moment (Variance)</h2>
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<ol>
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<li>The second moment of a probability distribution is the variance, which measures the spread or variability of the distribution around its mean.</li>
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<li>For a binomial distribution, the second moment (variance) is given by n * p * (1 - p), where n is the number of trials and p is the probability of success on each trial.</li>
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</ol>
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</body>
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‎Probability andStatistics/Prob Dist Functions/random_variables.html‎

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<title>Random Variables By Peerzada Aadil</title>
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<h1>Random Variables {Author: Peerzada Aadil}</h1>
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<h2>BHSS Sogam; Class:12th ; Subject:Statistics</h2>
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<p>A random variable is a variable that takes on different values based on the outcome of a random event. It is a fundamental concept in probability theory and statistics, used to quantify and analyze uncertainty. Random variables can be classified into two main types: discrete and continuous.</p>
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<ol>
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<li>Discrete Random Variables: These are random variables that can take on a countable number of distinct values.</li>
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<li>Continuous Random Variables: These are random variables that can take on an infinite number of values within a given range.</li>
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</ol>
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<h2>Examples of Discrete Random Variables</h2>
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<ol>
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<li>The number of heads obtained when flipping a coin multiple times.</li>
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<li>The number of cars passing through a toll booth in an hour.</li>
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<li>The number of students in a classroom.</li>
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<li>The number of goals scored in a soccer match.</li>
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<li>The number of phone calls received at a call center in a day.</li>
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<li>The number of students who attend a particular lecture.</li>
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<li>The number of teachers present in a school.</li>
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<li>The number of books available in a library on a given day.</li>
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<li>The number of patients in a hospitalon a given day.</li>
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<li>The number of classes attended by Toiba and Afifa today.</li>
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</ol>
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<h2>Examples of Continuous Random Variables</h2>
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<ol>
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<li>The height of individuals.</li>
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<li>The time it takes to complete a task.</li>
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<li>The temperature in a city.</li>
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<li>The weight of a person.</li>
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<li>The amount of rainfall in a day.</li>
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<li>The speed of a car on a highway.</li>
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<li>The length of a piece of string.</li>
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<li>The time it takes for a computer to process a task.</li>
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<li>The amount of money spent on groceries in a week.</li>
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<li>The distance traveled by a person in a day.</li>
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</ol>
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<h2>Conclusion</h2>
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<p>Understanding random variables is essential for analyzing and interpreting data in various fields, including statistics, engineering, economics, and natural sciences. By classifying random variables as discrete or continuous, we can apply appropriate mathematical tools and techniques to model and analyze uncertainty effectively.</p>
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‎Probability andStatistics/Prob Dist Functions/style.css‎

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