CODEWITHSUNDEEP
rnormal-distributionprobability-distribution
RickPeck
RickPeck

Reputation: 185

Random Variables with Standard Normal Distribution Falling into Specific Intervals

I'm working on a task where we look at 12 independent and identically distributed random variables - each of which have standard normal distribution.

From that I understand we have a mean of 0 and sd of 1.

We then have an interval of (-1.644, 1.644)

To find the probability of a single random variable landing in this interval I write:

(pnorm(1.644, mean = 0, sd = 1, lower.tail=TRUE) - pnorm(-1.644, mean = 0, sd = 1, lower.tail=TRUE))

Which returns the Probability of 0.8998238

I'm able to find the probability of at least one of the 12 random variables landing outside of the interval (-1.644, 1.644) with the following:

PROB_1 = 1-(0.8998238^12))
#PROB_1 = 0.7182333

However - How would if find the probability of Exactly 2 random variables landing outside of the interval? I've attempted the following:

((12*11)/2)*((1-0.7182333)^2)*(0.7182333^10)

I'm sure I'm missing something here, and there is a much easier way to solve this.

Any help is much appreciated.

Upvotes: 1

Views: 90

Answers (1)

Vons
Vons

Reputation: 3325

You need the binomial coefficient

prob=pnorm(1.644, mean = 0, sd = 1, lower.tail=TRUE)-pnorm(-1.644, mean = 0, sd = 1, lower.tail=TRUE)
dbinom(2, 12, 1-prob)

prob^10 * (1-prob)^2 * choose(12, 2)

0.2304877

Upvotes: 1

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