12th Maths Monthly Test - 1 (Probability Distributions )

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 1 (Probability Distributions )-Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

The random variable X has the probability density function f(x) = \(\begin{cases}ax+b \quad0 < x < 1 \\0\quad otherwise\end {cases}\) and E(X) = \(\frac{7}{12}\)then a and b are respectively.
1 and \(\frac{1}{2}\)

\(\frac{1}{2}\)and 1

2 and 1

1 and 2
Consider a game where the player tosses a six sided fair die. H the face that comes up is 6, the player wins Rs.36, otherwise he loses Rs. k² , where k is the face that comes up k = {I, 2,3,4, 5}. The expected amount to win at this game in Rs is
\(\frac{19}{6}\)

-\(\frac{19}{6}\)

\(\frac{3}{2}\)

-\(\frac{3}{2}\)

A rod of length 2l is broken into two pieces at random. The probability density function of the shorter of the two pieces is f(x) = \(\begin {cases} \frac{2}{x^3}\space 0<x>l \\0 \space \space 1 ≤ x < 2l \end {cases}\)

\(\frac{l}{2},\frac{l^2}{3}\)
\(\frac{l}{2},\frac{l^2}{6 }\)
\({1},\frac{l^2}{12}\)
\(\frac{1}{2},\frac{l^2}{12}\)

If f(x) = \(\cases { 2x \quad 0\le x \le a \\ 0 \quad otherwise}\) is a probability density function of a random variable, then the value of a is

A computer salesperson knows from his past experience that he sells computers to one in every twenty customers who enter the showroom. What is the probability that he will sell a computer to exactly two of the next three customers?

\(\frac{57}{20^3}\)
\(\frac{57}{20^2}\)
\(\frac{19^3}{20^3}\)
\(\frac{57}{20}\)

Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. Then the possible values of X are

i + 2n, i = 0,1,2... n
2i- n, i = 0,1,2... n
n - i, i = 0,1,2... n
2i + 2n, i = 0, 1, 2...n

If P{X = 0} = 1- P{X = I}. IfE[X) = 3Var(X), then P{X = 0}.

\(\frac{2}{3}\)
\(\frac{2}{5}\)
\(\frac{1}{5}\)
\(\frac{1}{3}\)

On a multiple-choice exam with 3 possible destructives for each of the 5 questions, the probability that a student will get 4 or more correct answers just by guessing is

\(\frac{11}{243}\)
\(\frac{3}{8}\)
\(\frac{1}{243}\)
\(\frac{5}{243}\)

Let X be random variable with probability density function f(x) = \(\begin{cases} \frac { 2 }{ { x }^{ 3 } } \quad 0\quad <\quad x\quad \ge \quad l \\ 0\quad \quad 1\quad \ge \quad x\quad <\quad 2l \end{cases}\) Which of the following statement is correct

both mean and variance exist
mean exists but variance does not exist
both mean and variance do not exist
variance exists but Mean does not exist

Suppose that X takes on one of the values 0, 1, and 2. If for some constant k, P(X = I) = k P(X = i-I) i = 1, 2 and P(X = 0) = \(\frac{1}{7}\)then the value of k is

Which of the following is a discrete random variable?
I. The number of cars crossing a particular signal in a day
II.The number of customers in a queue-to buy train tickets at a moment.
III.The time taken to complete a telephone call.

I and II
II only
III only
II and III

A pair of dice numbered 1, 2, 3, 4, 5, 6 of a six-sided die and 1, 2, 3, 4 of a four-sided die is rolled and the sum is determined. Let the random variable X denote this sum. Then the number of elements in the inverse image of 7 is

A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is

If in 6 trials, X is a binomial variate which foUows the relation 9P(X = 4) = P(X = 2), then the probability of success is

0.125
0.25
0.375
0.75

Four buses carrying 160 students from the same school arrive at a football stadium. The buses carry, respectively, 42, 36, 34, and 48 students. One of the students is randomly selected. Let X denote the number of students that were on the bus carrying the randomly selected student One of the 4 bus drivers is also randomly selected. Let Y denote the number of students on that bus. Then E[X] and E[Y] respectively are

50,40
40,50
40.75,40
41,41

The probability mass function of a random variable is defined as:

X

-2

-1

0

1

2

f(X)

K

2k

3k

4k

5k

\(\frac{1}{15}\)
\(\frac{1}{10}\)
\(\frac{1}{3}\)
\(\frac{2}{3}\)

Part - B

Generic 2 - Answer All Question

When do we say that a discrete random variable X is a binomial random variable.

Define discrete random variable

A coin is tossed until a head appears or the tail appears 4 times in succession. Find the probability
distribution of the number of tosses.

A random variable X has the following probability mass function.

X

1

2

3

4

5

f(X)

K²

2k²

3k²

2k

3k

Compute P(X = k)for the binomial distribution, B(n,p) where P(X = 10) = \(\begin{pmatrix} 10\\ 4\end{pmatrix}\)\(\begin{pmatrix} \frac{1}{5}\end{pmatrix}^4\)\(\begin{pmatrix}1- \frac{1}{5}\end{pmatrix}^{10-4}\)
Prove that E(aX+b)=aE(X)+b

If μ and σ² are the mean and variance of the discrete random variable X, and E(X + 3)=10 and E(X +
30)² = 116, find μ and σ² .

Find the variance of the binomial distribution with parameters 8 and \(\frac{1}{4}\)

Using binomial distribution find the mean and variance of X for the following experiments
(i) A fair coin is tossed 100 times, and X denote the number of heads.
(ii) A fair die is tossed 240 times, and X denote the number of times that four appeared.

Prove that Var(X)=E(X²) if E(X)=0

Is it possible that the mean of a binomial distribution is 15 and its standard deviation is 5?

How many types of random variables are there? What are they?

What is meant by the expected value of a random variable X?

Define Probability Density function

The probability density function of X is given by f(x) = \(\begin{cases}kxe^ {−2x}\quad forx > 0\\0\quad for x ≤ 0\end{cases}\)
Prove that Var(x)=E(X²)-{E(X)]²

The probability that a certain kind of component will survive a electrical test is \(\frac{3}{4}\).Find the probability that exactly 3 of the 5 components tested survive.

Part - C

Generic 3 - Answer All Question

A pair of fair dice is rolled once. Find the probability mass function to get the number of fours.

Two cards are drawn simultaneously from a well shuffled pack of 52 cards. Find the probability
distribution of number of jacks.

Four fair coins are tossed once. Find the probability mass function, mean and variance for number of
heads occurred.
Out of a group of 60 architects 40 are qualified and co-operative while the remaining are qualified
but remain reserved. Two architects are selected from the group at random. Find the probability
distribution of the number of architects who are qualified and co-operative. Which of the two values,
namely co-cooperativeness or reservedness, mentioned above, do you prefer and why?

The probability of a shooter hitting a target is \(\frac{3}{4}\).How many minimum number of times must he fire
so that the probability of hitting the target atleast once is more than 0.99?

Two balls are chosen randomly from an urn containing 6 red and 8 black balls. Suppose that we win
Rs. 15 for each red ball selected and we lose Rs. 10 for each black ball selected. X denotes the
winning amount, then find the values of X and number of points in its inverse images.

A person tosses a coin and is to receive Rs.4 for a head and has to pay Rs.2 for a tail. Find the
variance of the game.

Three fair coins are tossed simultaneously. Find the probability mass function for number of heads
occurred

A random variable X has the following probability distribution.

X_i

-2

-1

0

1

2

3

P_i

0.1

k

0.2

2k

0.3

k

i) find k
ii) find the mean of the distribution

In a game, a man wins Rs.5 for getting a number greater than 4 and loses Re.1 otherwise when a fair
dice is thrown. The man decided to throw a die thrice but to quit as and when he gets a number
greater than 4. Find the expected value of the amount he wins/loses

For 6 trials of an experiment, let X be a binomial variate which satisfies the relation
9P(X=4)=P(X=2).Find p.

Part - D

Generic 5 - Answer All Question

Ten coins are tossed simultaneously. What is the probability of getting (a) exactly 6 heads (b) at least
6 heads (c) at most 6 heads?

The time to failure in thousands of hours of an electronic equipment used in a manufactured
computer has the density function f(x) = \(\begin{cases}3e ^{ 3x}\quad x > 0\\0 \quad elsewhere \end{cases}\) Find the expected life of this electronic equipment.

From a lot of 10 items containing 3 defective items, 4 items are drawn at random. Find the mean and
variance of the number of defective items drawn.

Two balls are chosen randomly from an urn containing 8 white and 4 black balls. Suppose that we
win Rs 20 for each black ball selected and we lose Rs10 for each white ball selected. Find the
expected winning amount and variance
The mean and variance of a binomial distribution are 4 and 2 respectively. Find the probability of at
least 6 success.

The probability distribution of a random variable X is given by

X

0

1

2

3

P(X)

0.1

0.3

0.5

0.1

If Y=X² +3X, find the mean and the variance of Y.

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12th Maths Monthly Test - 1 (Probability Distributions )

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 1 (Probability Distributions )-Aug 2020

Part - A

Multiple Choice Question - Answer All Question

The random variable X has the probability density function f(x) = \(\begin{cases}ax+b \quad0 < x < 1 \\0\quad otherwise\end {cases}\) and E(X) = \(\frac{7}{12}\)then a and b are respectively.

Consider a game where the player tosses a six sided fair die. H the face that comes up is 6, the player wins Rs.36, otherwise he loses Rs. k2 , where k is the face that comes up k = {I, 2,3,4, 5}. The expected amount to win at this game in Rs is

A rod of length 2l is broken into two pieces at random. The probability density function of the shorter of the two pieces is f(x) = \(\begin {cases} \frac{2}{x^3}\space 0<x>l \\0 \space \space 1 ≤ x < 2l \end {cases}\)

If f(x) = \(\cases { 2x \quad 0\le x \le a \\ 0 \quad otherwise}\) is a probability density function of a random variable, then the value of a is

A computer salesperson knows from his past experience that he sells computers to one in every twenty customers who enter the showroom. What is the probability that he will sell a computer to exactly two of the next three customers?

Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. Then the possible values of X are

If P{X = 0} = 1- P{X = I}. IfE[X) = 3Var(X), then P{X = 0}.

On a multiple-choice exam with 3 possible destructives for each of the 5 questions, the probability that a student will get 4 or more correct answers just by guessing is

Let X be random variable with probability density function f(x) = \(\begin{cases} \frac { 2 }{ { x }^{ 3 } } \quad 0\quad <\quad x\quad \ge \quad l \\ 0\quad \quad 1\quad \ge \quad x\quad <\quad 2l \end{cases}\) Which of the following statement is correct

Suppose that X takes on one of the values 0, 1, and 2. If for some constant k, P(X = I) = k P(X = i-I) i = 1, 2 and P(X = 0) = \(\frac{1}{7}\)then the value of k is

Which of the following is a discrete random variable? I. The number of cars crossing a particular signal in a day II.The number of customers in a queue-to buy train tickets at a moment. III.The time taken to complete a telephone call.

A pair of dice numbered 1, 2, 3, 4, 5, 6 of a six-sided die and 1, 2, 3, 4 of a four-sided die is rolled and the sum is determined. Let the random variable X denote this sum. Then the number of elements in the inverse image of 7 is

A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is

If in 6 trials, X is a binomial variate which foUows the relation 9P(X = 4) = P(X = 2), then the probability of success is

The probability mass function of a random variable is defined as: X -2 -1 0 1 2 f(X) K 2k 3k 4k 5k

Part - B

Generic 2 - Answer All Question

When do we say that a discrete random variable X is a binomial random variable.

Define discrete random variable

A coin is tossed until a head appears or the tail appears 4 times in succession. Find the probability distribution of the number of tosses.

A random variable X has the following probability mass function. X 1 2 3 4 5 f(X) K2 2k2 3k2 2k 3k

Compute P(X = k)for the binomial distribution, B(n,p) where P(X = 10) = \(\begin{pmatrix} 10\\ 4\end{pmatrix}\)\(\begin{pmatrix} \frac{1}{5}\end{pmatrix}^4\)\(\begin{pmatrix}1- \frac{1}{5}\end{pmatrix}^{10-4}\)

Prove that E(aX+b)=aE(X)+b

If μ and σ2 are the mean and variance of the discrete random variable X, and E(X + 3)=10 and E(X + 30)2 = 116, find μ and σ2 .

Find the variance of the binomial distribution with parameters 8 and \(\frac{1}{4}\)

Using binomial distribution find the mean and variance of X for the following experiments (i) A fair coin is tossed 100 times, and X denote the number of heads. (ii) A fair die is tossed 240 times, and X denote the number of times that four appeared.

Prove that Var(X)=E(X2) if E(X)=0

Is it possible that the mean of a binomial distribution is 15 and its standard deviation is 5?

How many types of random variables are there? What are they?

What is meant by the expected value of a random variable X?

Define Probability Density function

The probability density function of X is given by f(x) = \(\begin{cases}kxe^ {−2x}\quad forx > 0\\0\quad for x ≤ 0\end{cases}\)

Prove that Var(x)=E(X2)-{E(X)]2

The probability that a certain kind of component will survive a electrical test is \(\frac{3}{4}\).Find the probability that exactly 3 of the 5 components tested survive.

Part - C

Generic 3 - Answer All Question

A pair of fair dice is rolled once. Find the probability mass function to get the number of fours.

Two cards are drawn simultaneously from a well shuffled pack of 52 cards. Find the probability distribution of number of jacks.

Four fair coins are tossed once. Find the probability mass function, mean and variance for number of heads occurred.

The probability of a shooter hitting a target is \(\frac{3}{4}\).How many minimum number of times must he fire so that the probability of hitting the target atleast once is more than 0.99?

Two balls are chosen randomly from an urn containing 6 red and 8 black balls. Suppose that we win Rs. 15 for each red ball selected and we lose Rs. 10 for each black ball selected. X denotes the winning amount, then find the values of X and number of points in its inverse images.

A person tosses a coin and is to receive Rs.4 for a head and has to pay Rs.2 for a tail. Find the variance of the game.

Three fair coins are tossed simultaneously. Find the probability mass function for number of heads occurred

A random variable X has the following probability distribution. Xi -2 -1 0 1 2 3 Pi 0.1 k 0.2 2k 0.3 k i) find k ii) find the mean of the distribution

In a game, a man wins Rs.5 for getting a number greater than 4 and loses Re.1 otherwise when a fair dice is thrown. The man decided to throw a die thrice but to quit as and when he gets a number greater than 4. Find the expected value of the amount he wins/loses

For 6 trials of an experiment, let X be a binomial variate which satisfies the relation 9P(X=4)=P(X=2).Find p.

Part - D

Generic 5 - Answer All Question

Ten coins are tossed simultaneously. What is the probability of getting (a) exactly 6 heads (b) at least 6 heads (c) at most 6 heads?

The time to failure in thousands of hours of an electronic equipment used in a manufactured computer has the density function f(x) = \(\begin{cases}3e ^{ 3x}\quad x > 0\\0 \quad elsewhere \end{cases}\) Find the expected life of this electronic equipment.

From a lot of 10 items containing 3 defective items, 4 items are drawn at random. Find the mean and variance of the number of defective items drawn.

Two balls are chosen randomly from an urn containing 8 white and 4 black balls. Suppose that we win Rs 20 for each black ball selected and we lose Rs10 for each white ball selected. Find the expected winning amount and variance

The mean and variance of a binomial distribution are 4 and 2 respectively. Find the probability of at least 6 success.

The probability distribution of a random variable X is given by X 0 1 2 3 P(X) 0.1 0.3 0.5 0.1 If Y=X2 +3X, find the mean and the variance of Y.

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12th Maths Monthly Test - 2 ( Discrete Mathematics

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12th Maths Monthly Test - 1 ( Applications of Vect

Consider a game where the player tosses a six sided fair die. H the face that comes up is 6, the player wins Rs.36, otherwise he loses Rs. k² , where k is the face that comes up k = {I, 2,3,4, 5}. The expected amount to win at this game in Rs is

Which of the following is a discrete random variable?
I. The number of cars crossing a particular signal in a day
II.The number of customers in a queue-to buy train tickets at a moment.
III.The time taken to complete a telephone call.

The probability mass function of a random variable is defined as:

X

-2

-1

0

1

2

f(X)

K

2k

3k

4k

5k

A coin is tossed until a head appears or the tail appears 4 times in succession. Find the probability
distribution of the number of tosses.

A random variable X has the following probability mass function.

X

1

2

3

4

5

f(X)

K²

2k²

3k²

2k

3k

If μ and σ² are the mean and variance of the discrete random variable X, and E(X + 3)=10 and E(X +
30)² = 116, find μ and σ² .

Using binomial distribution find the mean and variance of X for the following experiments
(i) A fair coin is tossed 100 times, and X denote the number of heads.
(ii) A fair die is tossed 240 times, and X denote the number of times that four appeared.

Prove that Var(X)=E(X²) if E(X)=0

Prove that Var(x)=E(X²)-{E(X)]²

Two cards are drawn simultaneously from a well shuffled pack of 52 cards. Find the probability
distribution of number of jacks.

Four fair coins are tossed once. Find the probability mass function, mean and variance for number of
heads occurred.

The probability of a shooter hitting a target is \(\frac{3}{4}\).How many minimum number of times must he fire
so that the probability of hitting the target atleast once is more than 0.99?

Two balls are chosen randomly from an urn containing 6 red and 8 black balls. Suppose that we win
Rs. 15 for each red ball selected and we lose Rs. 10 for each black ball selected. X denotes the
winning amount, then find the values of X and number of points in its inverse images.

A person tosses a coin and is to receive Rs.4 for a head and has to pay Rs.2 for a tail. Find the
variance of the game.

Three fair coins are tossed simultaneously. Find the probability mass function for number of heads
occurred

A random variable X has the following probability distribution.

X_i

-2

-1

0

1

2

3

P_i

0.1

k

0.2

2k

0.3

k

i) find k
ii) find the mean of the distribution

In a game, a man wins Rs.5 for getting a number greater than 4 and loses Re.1 otherwise when a fair
dice is thrown. The man decided to throw a die thrice but to quit as and when he gets a number
greater than 4. Find the expected value of the amount he wins/loses

For 6 trials of an experiment, let X be a binomial variate which satisfies the relation
9P(X=4)=P(X=2).Find p.

Ten coins are tossed simultaneously. What is the probability of getting (a) exactly 6 heads (b) at least
6 heads (c) at most 6 heads?

The time to failure in thousands of hours of an electronic equipment used in a manufactured
computer has the density function f(x) = \(\begin{cases}3e ^{ 3x}\quad x > 0\\0 \quad elsewhere \end{cases}\) Find the expected life of this electronic equipment.

From a lot of 10 items containing 3 defective items, 4 items are drawn at random. Find the mean and
variance of the number of defective items drawn.

Two balls are chosen randomly from an urn containing 8 white and 4 black balls. Suppose that we
win Rs 20 for each black ball selected and we lose Rs10 for each white ball selected. Find the
expected winning amount and variance

The mean and variance of a binomial distribution are 4 and 2 respectively. Find the probability of at
least 6 success.

The probability distribution of a random variable X is given by

X

0

1

2

3

P(X)

0.1

0.3

0.5

0.1

If Y=X² +3X, find the mean and the variance of Y.