Probability Distributions (p-2) Maths (commerce) - Probability Distributions (p-2)

If E(X) = m and Var(X) = m then X follows __________

The expected value of the sum of two numbers obtained when two fair dice are rolled is __________

If r.v. X assumes the values 1, 2, 3,………, 9 with equal probabilities, E(X) = 5.

If f(x) = kx (1 – x) for 0 < x < 1 = 0 otherwise then k = 12

In Binomial distribution, if n is very large and probability success of p is very small such that np = m (constant) then ________ distribution is applied.

In Binomial distribution probability of success ________ from trial to trial.

E(X) is considered to be __________ of the probability distribution of X.

Identify the random variable as discrete or continuous in each of the following. Identify its range if it is discrete.
A highway safety group is interested in the speed (km/hrs) of a car at a checkpoint.

Identify the random variable as discrete or continuous in each of the following. Identify its range if it is discrete.
Twelve of 20 white rats available for an experiment are male. A scientist randomly selects 5 rats and counts the number of female rats among them.

Identify the random variable as discrete or continuous in each of the following. Identify its range if it is discrete.
A person on a high protein diet is interested in the weight gained in a week.

Identify the random variable as discrete or continuous in each of the following. Identify its range if it is discrete.
Amount of syrup prescribed by a physician.

The following is the c.d.f. of a r.v. X.

X	-3	-2	-1	0	1	2	3	4
F(x)	0.1	0.3	0.5	0.65	0.75	0.85	0.9	1

Find the probability distribution of X and P(-1 ≤ X ≤ 2).

A fair coin is tossed 4 times. Let X denote the number of heads obtained. Identify the probability distribution of X and state the formula for p.m.f. of X.

Following is the probability distribution of an r.v. X.

X	-3	-2	-1	0	1	2	3
P(X=x)	0.05	0.1	0.15	0.20	0.25	0.15	0.1

Find the probability that
(i) X is positive.
(ii) X is non-negative.
(iii) X is odd.
(iv) X is even.

The probability that a machine will produce all bolts in a production run with in the specification is 0.9. A sample of 3 machines is taken at random. Calculate the probability that all machines will produce all bolts in a production run within the specification.

A large chain retailer purchases an electric device from the manufacturer. The manufacturer indicates that the defective rate of the device is 10%. The inspector of the retailer randomly selects 4 items from a shipment. Find the probability that the inspector finds at most one defective item in the 4 selected items.

An examination consists of 5 multiple choice questions, in each of which the candidate has to decide which one of 4 suggested answers is correct. A completely unprepared student guesses each answer randomly. Find the probability that this student gets 4 or more correct answers.

It is observed that it rains on 10 days out of 30 days. Find the probability that
(i) it rains on exactly 3 days of a week.
(ii) it rains at most 2 days a week.

In a large school, 80% of the students like mathematics. A visitor asks each of 4 students, selected at random, whether they like mathematics, (i) Calculate the probabilities of obtaining an answer yes from all of the selected students, (ii) Find the probability that the visitor obtains the answer yes from at least 3 students.

Find the probability of the number of successes in two tosses of a die, where success is defined as (i) number greater than 4 (ii) six appearing in at least one toss.

Given that X ~ B(n, p),
(i) if n = 10 and p = 0.4, find E(X) and Var(X).
(ii) if p = 0.6 and E(X) = 6, find n and Var(X).
(iii) if n = 25, E(X) = 10, find p and Var(X).
(iv) if n = 10, E(X) = 8, find Var(X).

In the following probability distribution of an r.v. X

X	1	2	3	4	5
P(X)	1/20	3/20	a	2a	1/20

Find a and obtain the c.d.f. of X.

A sample of 4 bulbs is drawn at random with replacement from a lot of 30 bulbs which includes 6 defective bulbs. Find the probability distribution of the number of defective bulbs.