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What is the first quarter quartile of having the following probability density function function f x is equals to 1 upon -72 by under root 3 x minus 10 to the power 2 upon Mein 72?
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What is the first quarter quartile of having the following probability...
First Quarter Quartile of Probability Density Function
Definition of Probability Density Function
A probability density function (PDF) is a function that describes the likelihood of obtaining a particular value from a continuous random variable. It is a non-negative function that integrates to 1 over its range.
Given PDF Function
f(x) = 1/(-72√3x-10)^2/72
Definition of First Quarter Quartile
The first quarter quartile (Q1) is the value that defines the point at which 25% of the observations lie below it, and 75% of the observations lie above it.
Calculating Q1
- Calculate the cumulative distribution function (CDF) of the given PDF.
- Use the CDF to determine the value of x that corresponds to the 25th percentile.
- Round the value of x to the nearest hundredth to obtain Q1.
Cumulative Distribution Function
The cumulative distribution function (CDF) of a continuous random variable X is defined as:
F(x) = P(X ≤ x)
Integrating the given PDF function from negative infinity to x:
F(x) = ∫(-∞,x) f(t) dt = ∫(-∞,x) 1/(-72√3t-10)^2/72 dt
After solving the integral, we get:
F(x) = (1/2) - (1/2) erf((√3x+10)/(6√2))
Finding the 25th Percentile
We need to find the value of x that corresponds to the 25th percentile, which is the point at which 25% of the observations lie below it. We can use the inverse of the CDF to find this value.
F(x) = 0.25
Substituting the value of F(x) in the CDF equation:
0.25 = (1/2) - (1/2) erf((√3x+10)/(6√2))
After solving for x, we get:
x = -1.12
Calculating Q1
Rounding the value of x to the nearest hundredth, we get:
Q1 = -1.12
Conclusion
The first quarter quartile of the given probability density function is -1.12.
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What is the first quarter quartile of having the following probability density function function f x is equals to 1 upon -72 by under root 3 x minus 10 to the power 2 upon Mein 72?
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What is the first quarter quartile of having the following probability density function function f x is equals to 1 upon -72 by under root 3 x minus 10 to the power 2 upon Mein 72? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about What is the first quarter quartile of having the following probability density function function f x is equals to 1 upon -72 by under root 3 x minus 10 to the power 2 upon Mein 72? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is the first quarter quartile of having the following probability density function function f x is equals to 1 upon -72 by under root 3 x minus 10 to the power 2 upon Mein 72?.
What is the first quarter quartile of having the following probability density function function f x is equals to 1 upon -72 by under root 3 x minus 10 to the power 2 upon Mein 72? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about What is the first quarter quartile of having the following probability density function function f x is equals to 1 upon -72 by under root 3 x minus 10 to the power 2 upon Mein 72? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is the first quarter quartile of having the following probability density function function f x is equals to 1 upon -72 by under root 3 x minus 10 to the power 2 upon Mein 72?.
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