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2. If X is a discrete random variable such that its deviations are small compared with its arithmetic mean ( ), so that ( ) and higher powers of ( ) may be neglected, then prove that Arithmetic mean( ), Geometric mean( ) and harmonic mean( ) of are in Geometric Progression (G.P). Further, prove that g^2 h^2 and prove gp?
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2. If X is a discrete random variable such that its deviations are sma...
Understanding the Means
The Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) of a discrete random variable X are defined as follows:
- Arithmetic Mean (AM): A = (x1 + x2 + ... + xn) / n
- Geometric Mean (GM): G = (x1 * x2 * ... * xn)^(1/n)
- Harmonic Mean (HM): H = n / (1/x1 + 1/x2 + ... + 1/xn)
When the deviations of X from its arithmetic mean are small, we can establish relationships between these means.
Proving G.M., A.M., and H.M. are in G.P.
1. Neglecting Higher Powers: Since the deviations are small, we assume that (xi - A) is negligible. Thus, we can state that:
- AM ≈ A
- GM ≈ A
- HM ≈ A
2. Using Inequalities: By the properties of means, we know that:
- AM ≥ GM ≥ HM
3. Establishing the Relationship: Given the above, we can express the relationship:
- (AM)(HM) = GM^2
This implies that AM, GM, and HM are in G.P.
Proving g^2 = h^2
To further analyze, we can define:
- g = GM
- h = HM
From the previous relationship:
- g^2 = (AM * HM)
Thus, we have:
- g^2 = h^2
This establishes that the squares of the geometric and harmonic means are equal, reinforcing that they exhibit a geometric progression.
Conclusion
In conclusion, for a discrete random variable X with small deviations, the Arithmetic Mean, Geometric Mean, and Harmonic Mean are in Geometric Progression. This relationship showcases the inherent link between different types of averages under specified conditions.
The Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) of a discrete random variable X are defined as follows:
- Arithmetic Mean (AM): A = (x1 + x2 + ... + xn) / n
- Geometric Mean (GM): G = (x1 * x2 * ... * xn)^(1/n)
- Harmonic Mean (HM): H = n / (1/x1 + 1/x2 + ... + 1/xn)
When the deviations of X from its arithmetic mean are small, we can establish relationships between these means.
Proving G.M., A.M., and H.M. are in G.P.
1. Neglecting Higher Powers: Since the deviations are small, we assume that (xi - A) is negligible. Thus, we can state that:
- AM ≈ A
- GM ≈ A
- HM ≈ A
2. Using Inequalities: By the properties of means, we know that:
- AM ≥ GM ≥ HM
3. Establishing the Relationship: Given the above, we can express the relationship:
- (AM)(HM) = GM^2
This implies that AM, GM, and HM are in G.P.
Proving g^2 = h^2
To further analyze, we can define:
- g = GM
- h = HM
From the previous relationship:
- g^2 = (AM * HM)
Thus, we have:
- g^2 = h^2
This establishes that the squares of the geometric and harmonic means are equal, reinforcing that they exhibit a geometric progression.
Conclusion
In conclusion, for a discrete random variable X with small deviations, the Arithmetic Mean, Geometric Mean, and Harmonic Mean are in Geometric Progression. This relationship showcases the inherent link between different types of averages under specified conditions.
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2. If X is a discrete random variable such that its deviations are small compared with its arithmetic mean ( ), so that ( ) and higher powers of ( ) may be neglected, then prove that Arithmetic mean( ), Geometric mean( ) and harmonic mean( ) of are in Geometric Progression (G.P). Further, prove that g^2 h^2 and prove gp? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about 2. If X is a discrete random variable such that its deviations are small compared with its arithmetic mean ( ), so that ( ) and higher powers of ( ) may be neglected, then prove that Arithmetic mean( ), Geometric mean( ) and harmonic mean( ) of are in Geometric Progression (G.P). Further, prove that g^2 h^2 and prove gp? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 2. If X is a discrete random variable such that its deviations are small compared with its arithmetic mean ( ), so that ( ) and higher powers of ( ) may be neglected, then prove that Arithmetic mean( ), Geometric mean( ) and harmonic mean( ) of are in Geometric Progression (G.P). Further, prove that g^2 h^2 and prove gp?.
2. If X is a discrete random variable such that its deviations are small compared with its arithmetic mean ( ), so that ( ) and higher powers of ( ) may be neglected, then prove that Arithmetic mean( ), Geometric mean( ) and harmonic mean( ) of are in Geometric Progression (G.P). Further, prove that g^2 h^2 and prove gp? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about 2. If X is a discrete random variable such that its deviations are small compared with its arithmetic mean ( ), so that ( ) and higher powers of ( ) may be neglected, then prove that Arithmetic mean( ), Geometric mean( ) and harmonic mean( ) of are in Geometric Progression (G.P). Further, prove that g^2 h^2 and prove gp? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 2. If X is a discrete random variable such that its deviations are small compared with its arithmetic mean ( ), so that ( ) and higher powers of ( ) may be neglected, then prove that Arithmetic mean( ), Geometric mean( ) and harmonic mean( ) of are in Geometric Progression (G.P). Further, prove that g^2 h^2 and prove gp?.
Solutions for 2. If X is a discrete random variable such that its deviations are small compared with its arithmetic mean ( ), so that ( ) and higher powers of ( ) may be neglected, then prove that Arithmetic mean( ), Geometric mean( ) and harmonic mean( ) of are in Geometric Progression (G.P). Further, prove that g^2 h^2 and prove gp? in English & in Hindi are available as part of our courses for Mathematics.
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