Question

Based on previous experience, the production manager knows that 5% of components produced in one production process have a defect. Let's take a sample of six components.

a)               The probability that a random variable takes a value less than 3 is: Odgovor

b)               The expected value of the random variable is: Odgovor

c)               The standard deviation of the random variable is:Odgovor

Answer 1

Given:

Probability of success = p = 0.05

Number of sample = n = 6

Let X be the random variable.

X ~ Binomial (n=6, p=0.05)

The probability density function of binomial distribution is given by

P(X=X) = (nCx) * p^x (1-p)^n-x

a) The probability that a random variable takes a value less than 3 :

P(X < 3) = P(X=0 + P(X=1) + P(X=2)

= (6Cx) * (0.05)^x (1-0.05)^6-x

= 0.9978

P(X < 3) = 0.9978

The probability that a random variable takes a value less than 3 is 0.9978

b) The expected value of the random variable:

E(X) = n*p = 6*0.05 = 0.3

c) The standard deviation of the random variable :

SD(X) = √np(1-p) = √6*0.05(1-0.05) = √0.3*0.95 = 0.534