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The probability density function of continuous random variable is defined as follows the perfect c when -1 is less than or equal to X1 less than or equal to is equal to zero otherwise the value of c is?
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The probability density function of continuous random variable is defi...
To determine the value of the constant \( c \) in the probability density function (PDF) of a continuous random variable, we first need to ensure that the total area under the PDF over its defined range equals 1.
Probability Density Function (PDF)
The PDF is defined as follows:
- \( f(x) = c \) for \( -1 \leq x \leq 1 \)
- \( f(x) = 0 \) otherwise
This means that \( c \) is a constant value between -1 and 1, and the PDF is zero outside this interval.
Condition for Probability Density Function
For \( f(x) \) to be a valid PDF, the following condition must be satisfied:
- \( \int_{-\infty}^{\infty} f(x) \, dx = 1 \)
Given our function:
- \( \int_{-1}^{1} c \, dx = 1 \)
Calculating the Integral
Carrying out the integration:
- \( \int_{-1}^{1} c \, dx = c \cdot (1 - (-1)) = c \cdot 2 = 2c \)
Setting this equal to 1 gives us:
- \( 2c = 1 \)
Solving for \( c \)
Now, we can solve for \( c \):
- \( c = \frac{1}{2} \)
Thus, the value of \( c \) that satisfies the condition of a probability density function is:
- \( c = \frac{1}{2} \)
Conclusion
Therefore, using this value of \( c \), the PDF becomes:
- \( f(x) = \frac{1}{2} \) for \( -1 \leq x \leq 1 \)
- \( f(x) = 0 \) otherwise
This ensures that the total area under the curve of the PDF is equal to 1, validating it as a proper probability density function.
Probability Density Function (PDF)
The PDF is defined as follows:
- \( f(x) = c \) for \( -1 \leq x \leq 1 \)
- \( f(x) = 0 \) otherwise
This means that \( c \) is a constant value between -1 and 1, and the PDF is zero outside this interval.
Condition for Probability Density Function
For \( f(x) \) to be a valid PDF, the following condition must be satisfied:
- \( \int_{-\infty}^{\infty} f(x) \, dx = 1 \)
Given our function:
- \( \int_{-1}^{1} c \, dx = 1 \)
Calculating the Integral
Carrying out the integration:
- \( \int_{-1}^{1} c \, dx = c \cdot (1 - (-1)) = c \cdot 2 = 2c \)
Setting this equal to 1 gives us:
- \( 2c = 1 \)
Solving for \( c \)
Now, we can solve for \( c \):
- \( c = \frac{1}{2} \)
Thus, the value of \( c \) that satisfies the condition of a probability density function is:
- \( c = \frac{1}{2} \)
Conclusion
Therefore, using this value of \( c \), the PDF becomes:
- \( f(x) = \frac{1}{2} \) for \( -1 \leq x \leq 1 \)
- \( f(x) = 0 \) otherwise
This ensures that the total area under the curve of the PDF is equal to 1, validating it as a proper probability density function.
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The probability density function of continuous random variable is defined as follows the perfect c when -1 is less than or equal to X1 less than or equal to is equal to zero otherwise the value of c is?
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The probability density function of continuous random variable is defined as follows the perfect c when -1 is less than or equal to X1 less than or equal to is equal to zero otherwise the value of c is? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The probability density function of continuous random variable is defined as follows the perfect c when -1 is less than or equal to X1 less than or equal to is equal to zero otherwise the value of c is? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The probability density function of continuous random variable is defined as follows the perfect c when -1 is less than or equal to X1 less than or equal to is equal to zero otherwise the value of c is?.
The probability density function of continuous random variable is defined as follows the perfect c when -1 is less than or equal to X1 less than or equal to is equal to zero otherwise the value of c is? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The probability density function of continuous random variable is defined as follows the perfect c when -1 is less than or equal to X1 less than or equal to is equal to zero otherwise the value of c is? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The probability density function of continuous random variable is defined as follows the perfect c when -1 is less than or equal to X1 less than or equal to is equal to zero otherwise the value of c is?.
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