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For a normal distribution having mean=2 and variance=4,Fourth central movement u4 is Ans 48?
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For a normal distribution having mean=2 and variance=4,Fourth central ...
Calculation of Fourth Central Moment (u4) for Normal Distribution
To calculate the fourth central moment (u4) for a normal distribution having mean=2 and variance=4, we can use the formula:
u4 = E[(X-μ)^4]
where, X is the random variable, μ is the mean, and E[.] denotes the expected value.
Step-by-Step Solution:
Therefore, the fourth central moment (u4) for the given normal distribution is 48.
To calculate the fourth central moment (u4) for a normal distribution having mean=2 and variance=4, we can use the formula:
u4 = E[(X-μ)^4]
where, X is the random variable, μ is the mean, and E[.] denotes the expected value.
Step-by-Step Solution:
- Firstly, we need to find the standard deviation of the normal distribution, which is the square root of variance. So, the standard deviation is sqrt(4) = 2.
- Next, we can use the formula for the fourth central moment:
u4 = E[(X-μ)^4]
= E[(X-2)^4] - As the distribution is normal, we know that the central moments of odd order are zero. So, we only need to consider the even order moments.
u4 = E[(X-2)^4]
= E[X^4 - 8X^3 + 24X^2 - 32X + 16]
= E[X^4] - 8E[X^3] + 24E[X^2] - 32E[X] + 16 - Now, we can use the properties of the standard normal distribution to find the values of E[X^4], E[X^3], E[X^2], and E[X].
E[X] = μ = 2
E[X^2] = μ^2 + σ^2 = 2^2 + 2^2 = 8
E[X^3] = 0 (as the distribution is symmetric)
E[X^4] = 3σ^4 + 6μ^2σ^2 + μ^4 = 3(2^4) + 6(2^2)(2^2) + 2^4 = 48 - Substituting these values in the formula for u4:
u4 = E[X^4] - 8E[X^3] + 24E[X^2] - 32E[X] + 16
= 48 - 0 + 24(8) - 32(2) + 16
= 48
Therefore, the fourth central moment (u4) for the given normal distribution is 48.
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For a normal distribution having mean=2 and variance=4,Fourth central ...
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For a normal distribution having mean=2 and variance=4,Fourth central movement u4 is Ans 48?
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For a normal distribution having mean=2 and variance=4,Fourth central movement u4 is Ans 48? 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 For a normal distribution having mean=2 and variance=4,Fourth central movement u4 is Ans 48? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For a normal distribution having mean=2 and variance=4,Fourth central movement u4 is Ans 48?.
For a normal distribution having mean=2 and variance=4,Fourth central movement u4 is Ans 48? 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 For a normal distribution having mean=2 and variance=4,Fourth central movement u4 is Ans 48? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For a normal distribution having mean=2 and variance=4,Fourth central movement u4 is Ans 48?.
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