If
𝑋
∼
𝑁
(
5
,
2
)
, then what code will calculate
𝑃
(
𝑋

1
)
?

Question

If 
𝑋
∼
𝑁
(
5
,
2
)
, then what code will calculate
𝑃
(
𝑋
>
1
)
?

BlackTom AI · Accepted Answer

We begin by restating the problem: X is distributed as N(5, 2) and we want P(X > 1).

First, clarify the notation: N(μ, σ^2) means the mean is μ and the variance is σ^2. Here the second parameter is 2, so the standard deviation is sqrt(2).

To compute P(X > 1) using the normal distribution, we can use the cumulative distribution function (CDF) via pnorm in R-like syntax. P(X > 1) = 1 - P(X ≤ 1) = 1 - pnorm(1, mean = 5, sd = sqrt(2)).

Option analysis:
- The expression 1 - pnorm(1, 5, sqrt(2)) exactly represents P(X > 1) for X ~ N(5, 2), since pnorm(1, 5, sqrt(2)) computes P(X ≤ 1) with the specified mean and standard deviation, and subtracting from 1 yields the tail probability.

Why this is correct: using the standard approach for a normal variable, convert the tail probability into the complement of the CDF at the point 1, applying the correct mean (5) and standard deviation (sqrt(2)). Numerically, z = (1 - 5) / sqrt(2) = -4 / 1.414... ≈ -2.828, so P(X > 1) ≈ Φ(2.828) ≈ 0.9977, which aligns with the complement calculation 1 - Φ(-2.828).

Potential confusion to avoid: if someone misinterprets the second parameter as the standard deviation, that would lead to an incorrect value only if the parameter usage differs from N(μ, σ^2). Here the expression uses sd = sqrt(2) explicitly, which is consistent with the standard deviation being the square root of the variance 2.

In summary, the given option 1-pnorm(1,5,sqrt(2)) is the correct formulation for P(X > 1) when X ~ N(5, 2).

DATA1001/1901 (ND) Evaluate Quiz 4

If 𝑋 ∼ 𝑁 ( 5 , 2 ) , then what code will calculate 𝑃 ( 𝑋 > 1 ) ?

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