Question

the sum of independent normally distributed random variables is normally distributed with mean equal to the sum of the individual means and variance equal to the sum of the individual variances. if x is the sum of three independent normally distributed random variables with respective means 100, 150, and 200 and respective standard deviations 15, 20, and 25, the probability that x is between 420 and 460 is closest to which of the following?

Answer 1

The probability that x is between 420 and 460 is 0.25778

The sum of independent normally distributed random variables is normally distributed with mean equal to the sum of the individual means and variance equal to the sum of the individual variances.

so, the mean would be,

μ = 100 + 150 + 200

μ = 450

and standard deviation would be,

σ = 15+ 20 + 25

σ = 60

We need to find the probability that x is between 420 and 460.

P(420 < x < 460)

= P( 420 -  μ < x -  μ < 460 -  μ)

= P((420 -  μ)/σ < (x - μ)/σ < (460 -  μ)/σ)

= P((420 -  450)/60 < Z < (460 -  450)/60)

= P (-1/2 < Z < 1/6)

= P(-0.5<x<0.167)

= 0.25778

Therefore, the probability is P(420 < x < 460) = 0.25778

Learn more about the probability here:

brainly.com/question/11234923

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