Balbharati solutions for मैथमेटिक्स एण्ड स्टैटिस्टिक्स २ (कॉमर्स) [अंग्रेजी] कक्षा १२ महाराष्ट्र स्टेट बोर्ड chapter 8 - Probability Distributions [Latest edition]

Let X represent the difference between the number of heads and the number of tails when a coin is tossed 6 times. What are the possible values of X?

An urn contains 5 red and 2 black balls. Two balls are drawn at random. X denotes number of black balls drawn. What are the possible values of X?

Determine whether each of the following is a probability distribution. Give reasons for your answer.

Determine whether each of the following is a probability distribution. Give reasons for your answer.

Determine whether each of the following is a probability distribution. Give reasons for your answer.

Determine whether each of the following is a probability distribution. Give reasons for your answer.

Determine whether each of the following is a probability distribution. Give reasons for your answer.

Determine whether each of the following is a probability distribution. Give reasons for your answer.

Find the probability distribution of number of heads in two tosses of a coin.

Find the probability distribution of number of number of tails in three tosses of a coin

Find the probability distribution of number of heads in four tosses of a coin

Find the probability distribution of the number of successes in two tosses of a die if success is defined as getting a number greater than 4.

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.

A coin is biased so that the head is 3 times as likely to occur as tail. Find the probability distribution of number of tails in two tosses.

A random variable X has the following probability distribution:

Find:

1. k
2. P(X < 3)
3. P(X > 4)

Find expected value and variance of X for the following p.m.f.

Find expected value and variance of X, the number on the uppermost face of a fair die.

Find the mean of the number of heads in three tosses of a fair coin.

Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X

Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers. Find E(X).

Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance of X.

A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 years. If X denotes the age of a randomly selected student, find the probability distribution of X. Find the mean and variance of X.

70% of the members favour and 30% oppose a proposal in a meeting. The random variable X takes the value 0 if a member opposes the proposal and the value 1 if a member is in favour. Find E(X) and Var(X).

Check whether the following is a p.d.f. 

f(x) = `{(x, "for"  0 ≤ x ≤ 1),(2 - x, "for"  1 < x ≤ 2.):}`

Check whether the following is a p.d.f.

f(x) = 2  for 0 < x < q.

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

f(x) = `{(x/(8),  "for"  0 < x < 4),(0,  "otherwise."):}`

Find P(x < 1.5)

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

f(x) = `{(x/(8),  "for"  0 < x < 4),(0,  "otherwise."):}`

Find P(1 < x < 2)

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

f(x) = `{(x/(8),  "for"  0 < x < 4),(0,  "otherwise."):}`

Find P(x > 2)

It is felt that error in measurement of reaction temperature (in celsius) in an experiment is a continuous r.v. with p.d.f.

f(x) = `{(x^3/(64),  "for"  0 ≤ x ≤ 4),(0,   "otherwise."):}`
Verify whether f(x) is a p.d.f.

It is felt that error in measurement of reaction temperature (in celsius) in an experiment is a continuous r.v. with p.d.f.

f(x) = `{(x^3/(64),  "for"  0 ≤ x ≤ 4),(0,   "otherwise."):}`
Find P(0 < X ≤ 1).

It is felt that error in measurement of reaction temperature (in celsius) in an experiment is a continuous r.v. with p.d.f.

f(x) = `{(x^3/(64),  "for"  0 ≤ x ≤ 4),(0,   "otherwise."):}`
Find probability that X is between 1 and 3..

Find k if the following function represents the p. d. f. of a r. v. X.

f(x) = `{(kx,  "for"  0 < x < 2),(0,  "otherwise."):}`

Also find `"P"[1/4 < "X" < 1/2]`

Find k, if the following function represents p.d.f. of r.v. X.

f(x) = kx(1 – x), for 0 < x < 1 and = 0, otherwise.

Also, find `P(1/4 < x < 1/2) and P(x < 1/2)`.

Let X be the amount of time for which a book is taken out of library by a randomly selected student and suppose that X has p.d.f.

f(x) = `{(0.5x,  "for" 0 ≤ x ≤ 2),(0,  "otherwise".):}`
Calculate : P(X ≤ 1)

Let X be the amount of time for which a book is taken out of library by a randomly selected student and suppose that X has p.d.f.

f(x) = `{(0.5x, "for" 0 ≤ x ≤ 2),(0, "otherwise".):}`
Calculate : P(0.5 ≤ X ≤ 1.5)

Let X be the amount of time for which a book is taken out of library by a randomly selected student and suppose that X has p.d.f.

f(x) = `{(0.5x, "for" 0 ≤ x ≤ 2),(0, "otherwise".):}`
Calculate : P(X ≥ 1.5)

Suppose X is the waiting time (in minutes) for a bus and its p. d. f. is given by

f(x) = `{(1/5,  "for"  0 ≤ x ≤ 5),(0,  "otherwise"):}`

Find the probability that waiting time is between 1 and 3 minutes.

Suppose X is the waiting time (in minutes) for a bus and its p. d. f. is given by

f(x) = `{(1/5,  "for"  0 ≤ x ≤ 5),(0,  "otherwise".):}`
Find the probability that waiting time is more than 4 minutes.

Suppose error involved in making a certain measurement is a continuous r. v. X with p.d.f.

f(x) = `{("k"(4 - x^2),  "for" -2 ≤ x ≤ 2),(0,  "otherwise".):}`
compute P(X > 0)

Suppose error involved in making a certain measurement is a continuous r. v. X with p.d.f.

f(x) = `{("k"(4 - x^2), "for" -2 ≤ x ≤ 2),(0, "otherwise".):}`
compute P(–1 < X < 1)

Suppose error involved in making a certain measurement is a continuous r. v. X with p.d.f.

f(x) = `{("k"(4 - x^2), "for" -2 ≤ x ≤ 2),(0, "otherwise".):}`
compute P(X < – 0.5 or X > 0.5)

Following is the p. d. f. of a continuous r.v. X.

f(x) = `{(x/8,  "for"  0 < x < 4),(0,  "otherwise".):}`
Find expression for the c.d.f. of X.

Following is the p. d. f. of a continuous r.v. X.

f(x) = `{(x/8,  "for"  0 < x < 4),(0,  "otherwise".):}`
Find F(x) at x = 0.5, 1.7 and 5.

The p.d.f. of a continuous r.v. X is

f(x) = `{((3x^2)/(8),  0 < x < 2),(0,   "otherwise".):}`
Determine the c.d.f. of X and hence find P(X < 1)

The p.d.f. of a continuous r.v. X is

f(x) = `{((3x^2)/(8), 0 < x < 2),(0, "otherwise".):}`
Determine the c.d.f. of X and hence find P(X < –2)

The p.d.f. of a continuous r.v. X is

f(x) = `{((3x^2)/(8),  0 < x < 2),(0, "otherwise".):}`
Determine the c.d.f. of X and hence find P(X > 0)

The p.d.f. of a continuous r.v. X is

f(x) = `{((3x^2)/(8), 0 < x < 2),(0, "otherwise".):}`
Determine the c.d.f. of X and hence find P(1 < X < 2)

If a r.v. X has p.d.f f(x) = `{("c"/x","  1 < x < 3"," "c" > 0),(0","  "otherwise"):}` 
Find c, E(X), and Var(X). Also Find F(x).

A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of 2 successes

A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of at least 3 successes

A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of at most 2 successes.

A pair of dice is thrown 3 times. If getting a doublet is considered a success, find the probability of two successes.

There are 10% defective items in a large bulk of items. What is the probability that a sample of 4 items will include not more than one defective item?

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards; find the probability that all the five cards are spades.

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards, find the probability that only 3 cards are spades

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards; find the probability that none is a spade.

The probability that a bulb produced by a factory will fuse after 200 days of use is 0.2. Let X denote the number of bulbs (out of 5) that fuse after 200 days of use. Find the probability of (i) X = 0, (ii) X ≤ 1, (iii) X > 1, (iv) X ≥ 1.

10 balls are marked with digits 0 to 9. If four balls are selected with replacement. What is the probability that none is marked 0?

In a multiple choice test with three possible answers for each of the five questions, what is the probability of a candidate getting four or more correct answers by random choice?

Find the probability of throwing at most 2 sixes in 6 throws of a single die.

Given that X ~ B(n, p), if n = 10 and p = 0.4, find E(X) and Var(X)

Given X ~ B(n, p) if p = 0.6 and E(X) = 6, find n and Var(X). 

Given that X ~ B(n,p), if n = 25, E(X) = 10, find p and Var (X).

Given that X ~ B(n, p), if n = 10, E(X) = 8, find Var(X).

If X has Poisson distribution with m = 1, then find P(X ≤ 1) given e⁻¹ = 0.3678

If X~P(0.5), then find P(X = 3) given e^−0.5 = 0.6065.

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

The number of complaints which a bank manager receives per day follows a Poisson distribution with parameter m = 4. Find the probability that the manager receives a) only two complaints on a given day, b) at most two complaints on a given day. Use e⁻⁴ = 0.0183.

A car firm has 2 cars, which are hired out day by day. The number of cars hired on a day follows Poisson distribution with mean 1.5. Find the probability that (i) no car is used on a given day, (ii) some demand is refused on a given day, given e^−1.5 = 0.2231.

Defects on plywood sheet occur at random with the average of one defect per 50 sq.ft. Find the probability that such a sheet has:

1. no defect
2. at least one defect
   Use e⁻¹ = 0.3678

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 exactly 5 rats inclusive. Given e^-5  = 0.0067.

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 more than 5 rats inclusive. Given e^-5  = 0.0067.

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.

F(x) is c.d.f. of discrete r.v. X whose p.m.f. is given by P(x) = `"k"^4C_x` , for x = 0, 1, 2, 3, 4 and P(x) = 0 otherwise then F(5) = _______

F(x) is c.d.f. of discrete r.v. X whose distribution is

Then F(– 3) = ______.

Choose the correct alternative :

X: is number obtained on upper most face when a fair die….thrown then E(X) = _______.

Choose the correct alternative :

If p.m.f. of r.v.X is given below.

Then Var(X) = _______

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

Given p.d.f. of a continuous r.v.X as

`f(x) = {{:(x^3/3",","for"-1 < x < 2), (0",","otherwise"):}`

then F(1) = _______.

X is r.v. with p.d.f. f(x) = `"k"/sqrt(x)`, 0 < x < 4 = 0 otherwise then x E(X) = _______

If X ∼ B`(20, 1/10)` then E(X) = ______.

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

If E(x) > Var(x) then X follows _______.

Fill in the blank :

The values of discrete r.v. are generally obtained by _______

Fill in the blank :

The value of continuous r.v. are generally obtained by _______

Fill in the blank :

If X is discrete random variable takes the value x₁, x₂, x₃,…, xn then \[\sum\limits_{i=1}^{n}\text{P}(x_i)\] = _______

If F(x) is the distribution function of discrete r.v.x with p.m.f. P(x) = `(x - 1)/(3)` for x = 1, 2, 3 and P(x) = 0 otherwise then F(4) = _______.

If F(x) is distribution function of discrete r.v.X with p.m.f. P(x) = `k^4C_x` for x = 0, 1, 2, 3, 4 and P(x) = 0 otherwise then F(–1) = _______

Fill in the blank :

E(x) is considered to be _______ of the probability distribution of x.

Fill in the blank :

If x is continuous r.v. and F(x_i) = P(X ≤ x_i) = `int_(-oo)^(oo) f(x)*dx` then F(x) is called _______

Fill in the blank :

In Binomial distribution probability of success Remains constant / independent from trial to trial.

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.

State whether the following is True or False :

If P(X = x) = `"k"[(4),(x)]` for x = 0, 1, 2, 3, 4 , then F(5) = `(1)/(4)` when F(x) is c.d.f.

State whether the following is True or False :

If F(x) is c.d.f. of discrete r.v. X then F(–3) = 0

State whether the following is True or False :

X is the number obtained on upper most face when a die is thrown then E(X) = 3.5.

State whether the following is True or False :

If p.m.f. of discrete r.v. X is

then E(x) = 2p.

State whether the following is True or False :

The p.m.f. of a r.v. X is P(x) = `(2x)/("n"("n" + 1))` , x = 1, 2, ……. n
= 0                  ,otherwise
Then E(x) = `(2"n" + 1)/(3)`

State whether the following is True or False :

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

State whether the following is True or False :

If X ~ B(n,p) and n = 6 and P(X = 4) = P(X = 2) then p = `(1)/(2)`

If r.v. X assumes values 1, 2, 3, ..., n with equal probabilities then E(X) = `(n + 1)/(2)`.

State whether the following is True or False :

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

Solve the following problem :

Identify the random variable as discrete or continuous in each of the following. Identify its range if it is discrete.

An economist is interested in knowing the number of unemployed graduates in the town with a population of 1 lakh.

Solve the following problem :

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.

Solve the following problem :

Identify the random variable as discrete or continuous in each of the following. Identify its range if it is discrete.

A person on high protein diet is interested in the weight gained in a week.

Solve the following problem :

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.

Solve the following problem :

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 check point.

Solve the following problem :

The probability distribution of a discrete r.v. X is as follows.

Determine the value of k.

Solve the following problem :

The probability distribution of a discrete r.v. X is as follows.

Find P(X ≤ 4), P(2 < X < 4), P(X ≤ 3).

Solve the following problem :

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

Find the probability that X is positive.

Solve the following problem :

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

Find the probability that X is non-negative

Solve the following problem:

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

Find the probability that X is odd.

Solve the following problem :

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

Find the probability that X is even.

Solve the following problem :

The p.m.f. of a r.v.X is given by

`P(X = x) = {(((5),(x)) 1/2^5", ", x = 0", "1", "2", "3", "4", "5.),(0,"otherwise"):}`

Show that P(X ≤ 2) = P(X ≤ 3).

Solve the following problem :

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

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

Solve the following problem :

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.

Solve the following problem :

Find the probability of the number of successes in two tosses of a die, where success is defined as number greater than 4.

Solve the following problem :

Find the probability of the number of successes in two tosses of a die, where success is defined as six appears in at least one toss.

A random variable X has the following probability distribution:

Find:

1. k
2. P(X < 3)
3. P(X > 4)

Solve the following problem :

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

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

Solve the following problem :

Find the expected value and variance of the r. v. X if its probability distribution is as follows.

Solve the following problem :

Find the expected value and variance of the r. v. X if its probability distribution is as follows.

Solve the following problem :

Find the expected value and variance of the r. v. X if its probability distribution is as follows.

Solve the following problem :

Find the expected value and variance of the r. v. X if its probability distribution is as follows.

Solve the following problem :

A player tosses two coins. He wins ₹ 10 if 2 heads appear, ₹ 5 if 1 head appears, and ₹ 2 if no head appears. Find the expected value and variance of winning amount.

Solve the following problem :

Let the p. m. f. of the r. v. X be

`"P"(x) = {((3 - x)/(10)", ","for"  x = -1", "0", "1", "2.),(0,"otherwise".):}`
Calculate E(X) and Var(X).

Solve the following problem :

Suppose error involved in making a certain measurement is a continuous r.v.X with p.d.f.

f(x) = `{("k"(4 - x^2), "for" -2 ≤ x ≤ 2),(0, "otherwise".):}`
Compute P(X > 0)

Solve the following problem :

Suppose error involved in making a certain measurement is a continuous r.v.X with p.d.f.

f(x) = `{("k"(4 - x^2), "for" -2 ≤ x ≤ 2),(0, "otherwise".):}`
Compute P(–1 < X < 1)

Solve the following problem :

Suppose error involved in making a certain measurement is a continuous r.v.X with p.d.f.

f(x) = `{("k"(4 - x^2), "for" -2 ≤ x ≤ 2),(0, "otherwise".):}`
Compute P(X < – 0.5 or X > 0.5)

Solve the following problem :

The p.d.f. of the r.v. X is given by

f(x) = `{((1)/(2"a")",", "for"  0 <  x= 2"a".),(0, "otherwise".):}`
Show that `"P"("X" < "a"/2) = "P"("X" > (3"a")/2)`

Solve the following problem :

Determine k if the p.d.f. of the r.v. is

f(x) = `{("ke"^(-thetax),  "for"  0 ≤ x < oo),(0, "otherwise".):}`
Find `"P"("X" > 1/theta)` and determine M is P(0 < X < M) = `(1)/(2)`

Solve the following problem :

The p.d.f. of the r.v. X is given by

f(x) = `{("k"/sqrt(x), "for"  0 < x < 4.),(0, "otherwise".):}`
Determine k, the c.d.f. of X, and hence find P(X ≤ 2) and P(X ≥ 1).

Solve the following problem :

Let X denote the reaction temperature in Celsius of a certain chemical process. Let X have the p. d. f.

f(x) = `{((1)/(10),  "for" -5 ≤ x < 5),(0, "otherwise".):}`
Compute P(X < 0).

Solve the following problem :

Let X ∼ B(10,0.2). Find P(X = 1)

Solve the following problem :

Let X ∼ B(10,0.2). Find P(X ≥1)

Solve the following problem :

Let X ∼ B(10,0.2). Find P(X ≤ 8).

Solve the following problem :

Let X∼B(n,p) If n = 10 and E(X)= 5, find p and Var(X).

Solve the following problem :

Let X∼B(n,p) If E(X) = 5 and Var(X) = 2.5, find n and p.

Solve the following problem :

If a fair coin is tossed 4 times, find the probability that it shows 3 heads

Solve the following problem :

If a fair coin is tossed 4 times, find the probability that it shows head in the first 2 tosses and tail in last 2 tosses.

Solve the following problem :

The probability that a bomb will hit the target is 0.8. Find the probability that, out of 5 bombs, exactly 2 will miss the target.

Solve the following problem :

The probability that a lamp in the classroom will burn is 0.3. 3 lamps are fitted in the classroom. The classroom is unusable if the number of lamps burning in it is less than 2. Find the probability that the classroom cannot be used on a random occasion.

Solve the following problem :

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.

Solve the following problem :

The probability that a component will survive a check test is 0.6. Find the probability that exactly 2 of the next 4 components tested survive.

Solve the following problem:

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 completely randomly. Find the probability that,

1. the student gets 4 or more correct answers.
2. the student gets less than 4 correct answers.

Solve the following problem :

The probability that a machine will produce all bolts in a production run within 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.

Solve the following problem :

A computer installation has 3 terminals. The probability that any one terminal requires attention during a week is 0.1, independent of other terminals. Find the probabilities that 0

Solve the following problem :

A computer installation has 3 terminals. The probability that any one terminal requires attention during a week is 0.1, independent of other terminals. Find the probabilities that 1 terminal requires attention during a week.

Solve the following problem :

In a large school, 80% of the students like mathematics. A visitor asks each of 4 students, selected at random, whether they like mathematics.

Calculate the probabilities of obtaining an answer yes from all of the selected students.

Solve the following problem :

In a large school, 80% of the students like mathematics. A visitor asks each of 4 students, selected at random, whether they like mathematics.

Find the probability that the visitor obtains the answer yes from at least 3 students.

Solve the following problem :

It is observed that it rains on 10 days out of 30 days. Find the probability that it rains on exactly 3 days of a week.

Solve the following problem :

It is observed that it rains on 10 days out of 30 days. Find the probability that it rains on at most 2 days of a week.

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.

Solve the following problem :

If X follows Poisson distribution with parameter m such that
`("P"("X" = x + 1))/("P"("X" = x)) = (2)/(x + 1)`
Find mean and variance of X.