8. (14; 2 for a), b), and c), 1 for d), 2 for e), and 5 for f)) Consider a rando

Question

8. (14; 2 for a), b), and c), 1 for d), 2 for e), and 5 for f)) Consider a random experiment consisting of a Bernouilli trial with a 10 % probability of success. Suppose as well that there is a series of 10 trials that we assume are independent of each other. This random experiment will give rise to a random variable that follows a binomial distribution with PDF p (X).

a) Calculate the value of p (10) – 10 successes in 10 trials.
b) Calculate the value of p (2) – 2 successes in 10 trials.
c) Calculate the value of p (0) – 0 successes in 10 trials.
d) Comment on any pattern that you see over the three previous points, and whether or not it makes sense.

e) Calculate the value of at least one success. (Hint: you have already almost answered this question, so there is a shortcut.)
f) Give the expected value, the variance, and the standard error of this random variable. At what probability value will there be the most dispersion?

Explanation / Answer

Modeling this as a Binomial distribution with the following parameters:

n = 10, p = 0.1

(a)

P(X=10) = 10C10*p10*(1-p)10-10 = 10C10*0.110*0.910-10 = 10-10

(b)

P(X=2) = 10C2*0.12*0.98 = 0.194

(c)

P(X=0) = 10C0*0.10*0.910 = 0.348

(d)

As the value of X decreases, the p-value increases. Yes it makes sense because probability is very small.

(e)

P(X>=1) = 1-P(X=0) = 1-0.348 = 0.652

Hope this helps !

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