Question

1. A Binomial random variable is an example of a, a continuous random variable b. a discrete random variable. c. a Binomial random variable is neither continuous nor discrete d. a Binomial random variable can be both continuous and discrete. Consider the following probability distribution where random variable X denotes the number of cups of coffee a random individual drinks in the morning P(x) 0.350 .400 .14 0.07 0.03 0.01 pe a. Compute the probability that a random individual drinks less than three cups of coffee in the morning 2pt) b. Compute P(X > 2). Spec. Compute the Mean number of cups of coffee a random person drinks in the morning. (pld . If two individuals are randomly selected, determine the probability that both drink 0 cups of coffee in the moming. Assume independence, Use the Binomial Distribution to answer the following: Your local grocery store has your favorite kind of ice cream in stock 65% of the time. You go to this grocery on 10 random occasions (You can assume independence). pt a . What is the probability that your favorite ice cream is in stock exactly 8 of the ten times?

Answer 1

Solution

Back-up Theory

If X ~ B(n, p). i.e., X has Binomial Distribution with parameters n and p, where

n = number of trials and p = probability of one success, then,

probability mass function (pmf) of X is given by p(x) = P(X = x) = (ⁿC_x)(p^x)(1 - p)^{n – x}..….......……..(1)

[This probability can also be directly obtained using Excel Function: Statistical, BINOMDIST]…….(1a)

Mean (average) of X = np…................................................................................................................(2)

Now to work out the solution,

Q 1

Option (b) Answer 1

[Because. Binomial variable can take values only at discrete points,

namely 0, 1, ...., n and cannot take any value in-between.]

Q 2

Part (a)

Probability a random individual drinks less than 3 cups

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

= 0.35 + 0.4 + 0.14

= 0.89 Answer 2

Part (b)

P(X > 2)

= 1 - P(X ≤ 2)

= 1 – 0.89 [vide Answer 2]

= 0.11 Answer 3

Part (c)

Mean = Σ(x = 0 to 5){x.P(x)}

= 1.06 Answer 4

Calculations

x	0	1	2	3	4	5	Total
P(x)	0.35	0.4	0.14	0.07	0.03	0.01	1
x.P(x)	0	0.4	0.28	0.21	0.12	0.05	1.06

Part (d)

Probability both of two randomly selected individuals drink zero cup of coffee

= (Probability one randomly selected individuals drinks zero cup of coffee)

[because of independence]

= 0.35²

= 0.1225 Answer 5

Q 3

Let X = number of times out of 10 times, the favourite ice cream is in stock.

Then, X ~ B(10, 0.65) ..........................................................................................................(3)

Part (a)

P(X = 8)

= 0.1757 [vide (3) and (1a)] Answer 6

Part (b)

P(X ≤ 2)

= 0.9140 [vide (3) and (1a)] Answer 7

Part (c)

P(X ≥ 5)

= 1 – 0.0949 [vide (3) and (1a)]

= 0.9051 Answer 8

Part (d)

Vide (2) and (3),

Average number of times the favourite ice cream is in stock = 10 x 0.65

= 6.5 Answer

DONE