Let x ∼ N(μ, σ²) If μ² = σ², (μ > 0), then the value of P(X < -μ | X < μ) in terms of cumulative function N (0, 1) is:

Concept:

The Standard Normal Distribution:

The standard normal distribution is a normal distribution with a Mean of 0 and a Standard Deviation of 1.

The standard normal distribution is centered at zero and the degree to which a given measurement deviates from the mean is given by standard deviation.

If X is normally distributed with mean μ and standard deviation σ, then \(Z = \frac{{X - μ }}{σ }\) is standard normally distributed with mean 0 and standard deviation 1.

Analysis:

Given: X ∼ N (μ, σ²)

μ² = σ² (μ > 0)

P(X < -μ | X < μ) = ?

Now,

Converting distribution of X ∼ N (μ, σ2) into normal distribution Z ∼ N (0, 1)

Where, \(Z=\dfrac{X-μ}{σ}\)

since, σ² = 1 we have, σ = 1

μ2 = σ2 and μ > 0 so μ = 1

When X  = -1,

\(Z=\dfrac{-1-1}{1}=-2\)

When X = 1, 

\(Z=\dfrac{1-1}{1}=0\)

P(X < -μ|X < μ) = P (X < -1 | X < 1)

= P (Z < -2 | Z < 0)

\(\rm P(Z<-2|Z<0)=\dfrac{P(Z<-2\cap Z<0)}{P(Z<0)}\)

= \(\rm \dfrac{P(Z<-2)}{P(Z<0)}=\dfrac{P(Z>2)}{P(Z<0)}=\dfrac{1-P(Z<2)}{1/2}\)

= 2 [1 - P(Z < 2)]

1. Normal distribution is symmetric about Y axis So P(X< -a) = P(X> a).

2.Total Area of the Normal distribution curve is 1. The half area lies on the left side of the mean and the half area lies on the right side of the mean.

3. For any random variable X: Half area lies on the left side of mean P(X > a) = 1- P(X < a).