{ \begin{tabular}{c|cccc} $x$ & 1 & 2 & 3 & 4 \\ \hline $P(X=x)$ & C & $c^2$ & $c^2 + C$ & $3c^2 + 2c$ \end{tabular} \}$(i) Obtain The Value Of ${$c\$}$.Hence, Rewrite The Probability Distribution.(ii) Obtain The Mean And Variance Of

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Introduction

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Probability distribution is a fundamental concept in statistics and probability theory. It is a function that describes the probability of each possible outcome in a random experiment. In this article, we will discuss a specific probability distribution and its properties.

Given Probability Distribution

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The given probability distribution is:

$x$	1	2	3	4
$P(X=x)$	$c$	$c^2$	$c^2 + c$	$3c^2 + 2c$

Obtaining the Value of $c$

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To obtain the value of $c$, we need to use the fact that the sum of probabilities of all possible outcomes is equal to 1.

$$\sum_{x=1}^{4} P(X=x) = 1$$

Substituting the given probabilities, we get:

$$c + c^2 + (c^2 + c) + (3c^2 + 2c) = 1$$

Simplifying the equation, we get:

$$5c^2 + 4c - 1 = 0$$

Solving the quadratic equation, we get:

$$c = \frac{-4 \pm \sqrt{16 + 20}}{10}$$

$$c = \frac{-4 \pm \sqrt{36}}{10}$$

$$c = \frac{-4 \pm 6}{10}$$

$$c = \frac{1}{5} \text{ or } c = -\frac{1}{5}$$

Since $c$ represents a probability, it must be a positive value. Therefore, we take $c = \frac{1}{5}$.

Rewriting the Probability Distribution

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Now that we have obtained the value of $c$, we can rewrite the probability distribution as:

$x$	1	2	3	4
$P(X=x)$	$\frac{1}{5}$	$\frac{1}{25}$	$\frac{6}{25}$	$\frac{18}{25}$

Obtaining the Mean

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The mean of a discrete random variable is given by:

$$\mu = \sum_{x} xP(X=x)$$

Substituting the values, we get:

$$\mu = 1 \times \frac{1}{5} + 2 \times \frac{1}{25} + 3 \times \frac{6}{25} + 4 \times \frac{18}{25}$$

Simplifying the expression, we get:

$$\mu = \frac{1}{5} + \frac{2}{25} + \frac{18}{25} + \frac{72}{25}$$

$$\mu = \frac{1 + 2 + 18 + 72}{25}$$

$$\mu = \frac{93}{25}$$

$$\mu = 3.72$$

Obtaining the Variance

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The variance of a discrete random variable is given by:

$$\sigma^2 = \sum_{x} (x - \mu)^2 P(X=x)$$

Substituting the values, we get:

$$\sigma^2 = (1 - 3.72)^2 \times \frac{1}{5} + (2 - 3.72)^2 \times \frac{1}{25} + (3 - 3.72)^2 \times \frac{6}{25} + (4 - 3.72)^2 \times \frac{18}{25}$$

Simplifying the expression, we get:

$$\sigma^2 = (2.72)^2 \times \frac{1}{5} + (1.72)^2 \times \frac{1}{25} + (0.28)^2 \times \frac{6}{25} + (0.28)^2 \times \frac{18}{25}$$

$$\sigma^2 = \frac{7.3944}{5} + \frac{2.9632}{25} + \frac{0.2496}{25} + \frac{0.2496}{25}$$

$$\sigma^2 = 1.46888 + 0.11824 + 0.009984 + 0.009984$$

$$\sigma^2 = 1.506056$$

$$\sigma^2 = 1.51$$

In conclusion, we have obtained the value of $c$ as $\frac{1}{5}$ and rewritten the probability distribution. We have also obtained the mean and variance of the random variable $X$.

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Introduction

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In our previous article, we discussed a specific probability distribution and its properties. In this article, we will answer some frequently asked questions related to probability distribution and its properties.

Q: What is a probability distribution?

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A: A probability distribution is a function that describes the probability of each possible outcome in a random experiment.

Q: What are the properties of a probability distribution?

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A: The properties of a probability distribution are:

• The sum of probabilities of all possible outcomes is equal to 1.
• The probability of each outcome is between 0 and 1.
• The probability of an outcome is non-negative.

Q: How do you calculate the mean of a probability distribution?

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A: The mean of a probability distribution is calculated using the formula:

$$\mu = \sum_{x} xP(X=x)$$

where $x$ is the value of the random variable and $P(X=x)$ is the probability of that value.

Q: How do you calculate the variance of a probability distribution?

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A: The variance of a probability distribution is calculated using the formula:

$$\sigma^2 = \sum_{x} (x - \mu)^2 P(X=x)$$

where $x$ is the value of the random variable, $\mu$ is the mean, and $P(X=x)$ is the probability of that value.

Q: What is the difference between mean and variance?

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A: The mean is a measure of the central tendency of a probability distribution, while the variance is a measure of the spread or dispersion of the distribution.

Q: How do you determine the type of probability distribution?

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A: The type of probability distribution can be determined by the shape of the distribution, the number of parameters, and the type of data. Some common types of probability distributions include:

• Discrete uniform distribution
• Continuous uniform distribution
• Binomial distribution
• Poisson distribution
• Normal distribution

Q: What is the importance of probability distribution in real-life applications?

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A: Probability distribution is used in many real-life applications, including:

• Insurance: to calculate the probability of an event occurring and the associated cost
• Finance: to calculate the probability of a stock price movement and the associated risk
• Engineering: to calculate the probability of a system failure and the associated risk
• Medicine: to calculate the probability of a disease occurring and the associated risk

Q: How do you interpret the results of a probability distribution?

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A: The results of a probability distribution can be interpreted in the following ways:

• The probability of an event occurring can be used to make decisions
• The mean and variance can be used to understand the central tendency and spread of the distribution
• The shape of the distribution can be used to understand the type of data and the associated risk

In conclusion, probability distribution is a fundamental concept in statistics and probability theory. It is used to describe the probability of each possible outcome in a random experiment. The properties of a probability distribution, such as the mean and variance, can be used to understand the central tendency and spread of the distribution. The type of probability distribution can be determined by the shape of the distribution, the number of parameters, and the type of data. Probability distribution is used in many real-life applications, including insurance, finance, engineering, and medicine.