HSC Commerce (English Medium) १२ वीं कक्षा - Maharashtra State Board Important Questions for Mathematics and Statistics

A job production unit has four jobs P, Q, R, and S which can be manufactured on each of the four machines I, II, III, and IV. The processing cost of each job for each machine is given in the following table:

Find the optimal assignment to minimize the total processing cost.

Five jobs are performed first on machine M₁ and then on machine M₂. Time taken in hours by each job on each machine is given below:

Determine the optimal sequence of jobs and total elapsed time. Also, find the idle time for two machines.

A department store has four workers to pack goods. The times (in minutes) required for each worker to complete the packings per item sold is given below. How should the manager of the store assign the jobs to the workers, so as to minimize the total time of packing?

A marketing manager has list of salesmen and territories. Considering the travelling cost of the salesmen and the nature of territory, the marketing manager estimates the total of cost per month (in thousand rupees) for each salesman in each territory. Suppose these amounts are as follows:

Find the assignment of salesman to territories that will result in minimum cost.

Six jobs are performed on Machines M₁ and M₂ respectively. Time in hours taken by each job on each machine is given below:

Determine the optimal sequence of jobs and find total elapsed time. Also find the idle time for machines M₁ and M₂.

Solution:

Given jobs can be arranged in optimal sequence as,

Total Elapsed time = `square` hrs.

Idle time for Machine M₁ = 43 – 42 = 1 hour.

Idle time for Machine M₂ = `square` hrs.

Three new machines M1, M2, M3 are to be installed in a machine shop. There are four vacant places A, B, C, D. Due to limited space, machine M2 can not be placed at B. The cost matrix (in hundred rupees) is as follows:

Determine the optimum assignment schedule and find the minimum cost.

The number of complaints which a bank manager receives per day is a Poisson random variable with parameter m = 4. Find the probability that the manager will receive -

(a) only two complaints on any given day.

(b) at most two complaints on any given day

[Use e^-4 =0.0183]

 If a random variable X follows Poisson distribution such that P(X = l) =P(X = 2), then find P(X ≥ 1).  [Use e^-2 = 0.1353] 

 If X has Poisson distribution with parameter m = 1, find P[X ≤ 1]  [Use `e^-1 = 0.367879`]

If X has a Poisson distribution with variance 2, find P (X = 4) 

[Use e^-2 = 0.1353] 

If X has a Poisson distribution with variance 2, find P(X ≤ 4) 

[Use e^-2 = 0.1353] 

If X has a Poisson distribution with variance 2, find 

Mean of X [Use e^-2 = 0.1353] 

If X has Poisson distribution with parameter m and P(X = 2) = P(X = 3), then find P(X ≥ 2). Use e⁻³ = 0.0497

It is known that, in a certain area of a large city, the average number of rats per bungalow is five. Assuming that the number of rats follows Poisson distribution, find the probability that a randomly selected bungalow has between 5 and 7 rats inclusive. Given e⁻⁵ = 0.0067.

If E(X) = m and Var(X) = m then X follows ______.

Solve the following problem :

If X follows Poisson distribution such that P(X = 1) = 0.4 and P(X = 2) = 0.2, find variance of X.

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

Choose the correct alternative:

A distance random variable X is said to have the Poisson distribution with parameter m if its p.m.f. is given by P(x) = `("e"^(-"m")"m"^"x")/("x"!)` the condition for m is ______

If X follows Poisson distribution such that P(X = 1) = 0.4 and P(X = 2) = 0.2, using the following activity find the value of m.

Solution: X : Follows Poisson distribution

∴ P(X) = `("e"^-"m" "m"^x)/(x!)`, P(X = 1) = 0.4 and P(X = 2) = 0.2

∴ P(X = 1) = `square` P(X = 2).

`("e"^-"m" "m"^x)/(1!) = square ("e"^-"m" "m"^2)/(2!)`,

`"e"^-"m" = square  "e"^-"m" "m"/2`, m ≠ 0

∴ m = `square`

State whether the following statement is true or false:

lf X ∼ P(m) with P(X = 1) = P(X = 2) then m = 1.