Let the mean and the standard deviation of the probability distribution
be μ and σ, respectively. If σ-μ=2, then σ+μ is equal to ________.

Solution

Maths sir · 5 steps

Write both μ and σ in terms of α, then square σ = μ + 2 to get an equation.

1. Find K. The probabilities must add to 1: 1/3 + K + 1/6 + 1/4 = 1. That gives K = 1 - 3/4 = 1/4.
2. Mean and variance. μ = α/3 + 1/4 + 0 - 3/4 = α/3 - 1/2 and E(X²) = α²/3 + 1/4 + 0 + 9/4 = α²/3 + 5/2. So σ² = E(X²) - μ² = 2α²/9 + α/3 + 9/4.
3. Use the condition. σ = μ + 2 = α/3 + 3/2. Square it and set it equal to the variance found above.

   2α²/9 + α/3 + 9/4 = (α/3 + 3/2)² = α²/9 + α + 9/4

4. Solve for alpha. Cancelling gives α²/9 = 2α/3, so α = 0 or α = 6. Reject α = 0, because X = 0 is already a separate value in the table, so the values would repeat.
5. So. With α = 6, μ = 2 - 1/2 = 3/2 and σ = μ + 2 = 7/2. Then σ + μ = 3/2 + 7/2 = 5.

The common slip: 1. Keeping α = 0 gives μ = -1/2 and σ = 3/2, so the sum is 1. But α = 0 repeats the value X = 0 already in the table, so that root must be rejected.

The idea to keep. Use Σ P = 1 first to fix the missing probability. Then σ² = E(X²) - μ². When a quadratic gives two roots, check that each root still gives a valid table.

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