JEE Main 2020 · 9 January · morning · Single correct · +4 / −1

Let the observations xᵢ (1 ≤ i ≤ 10) satisfy the
equations, Σᵢ₌₁¹⁰ ( x₁ - 5 ) = 10 and Σᵢ₌₁¹⁰ ( x₁ - 5 )² = 40.
If μ and λ are the mean and the variance of the
observations, x₁ – 3, x₂ – 3, ...., x₁₀ – 3, then
the ordered pair (μ, λ) is equal to :

  1. (a) (6, 6)
  2. (b) (3, 3)
  3. (c) (3, 6)
  4. (d) (6, 3)
Answer and solution

Answer (b) (3, 3)

Solution

Maths sir · 4 steps

Call xᵢ - 5 a new variable, find its mean and variance, then slide it by 2 to reach xᵢ - 3.

  1. Shift the data. The sums are meant to run over xᵢ, so read x₁ as xᵢ. Let yᵢ = xᵢ - 5. Then Σ yᵢ = 10 and Σ yᵢ² = 40, with 10 observations.
  2. Mean of y. The mean of yᵢ is 10/10 = 1, so the mean of xᵢ is 1 + 5 = 6.

    σ_y² = (Σ yᵢ²)/10 - ((Σ yᵢ)/10)² = 40/10 - 1² = 4 - 1 = 3

  3. Subtract 3 instead. Since xᵢ - 3 = yᵢ + 2, the new mean is μ = 1 + 2 = 3. Adding a constant to every observation moves the mean but does not change the spread, so λ = 3.
  4. So. (μ, λ) = (3, 3), which is option (b).

The idea to keep. Adding or subtracting a constant shifts the mean by that constant but leaves the variance unchanged. Work with whatever shifted variable the sums are given in.

Answer from the NTA final answer key. Solution by PrepWiser.

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Asked about 100 times in JEE Main since 2020. This question is from 2020.