12th Maths Monthly Test - 2 ( Probability Distributions)

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 2 ( Probability Distributions)-Aug 2020

Time : 2.30 Hours

Part - A

Generic 2 - Answer All Question

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.

Compute P(X = k)for the binomial distribution, B(n,p) where n=9,p =\(\frac{1}{2}\),k=7

Define discrete random variable

If the probability that a fluorescent light has a useful life of at least 600 hours is 0.9, find the
probabilities that among 12 such lights
(i) exactly 10 will have a useful life of at least 600 hours;
(ii) at least 11 will have a useful life of at I least 600 hours;
(iii) at least 2 will not have a useful life of at : least 600 hours.

In an investment a man can make a profit of Rs.5000 with a probability of 0.62 or a loss of Rs.8000
with a probability of 0.38. Find the expected gain or loss %

Suppose X is the number of tails occurred when three fair coins are tossed once simultaneously.
Find the values ofthe random variable X and number of points in its inverse images.

Define Distribution function.

Find the binomial distribution function for each of the following.
(i) Five fair coins are tossed once and X denotes the number of heads.
(ii) A fair die is rolled 10 times and X denotes the number of times 4 appeared.

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

Define Variance of a random variable X?

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.

A random variable X has the following probability mass function.

X

1

2

3

4

5

f(X)

K²

2k²

3k²

2k

3k

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.

What is the Probability Mass function of a binomial random variable?

For the random variable X with the given probability mass function as below, find the mean and
variance. \(f(x)=\begin{cases}\frac{4-x}{6}x=1,2,3 \end{cases}\)

A six sided die is marked '1' on one face, '3' on two of its faces, and '5' on remaining three faces. The
die is thrown twice. If X denotes the total score in two throws, find
(i) the probability mass function
(ii) the cumulative distribution function
(iii) P(4 ≤ X < 10)
(iv) P(X ≥ 6)

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 .

Define a random variable

Compute P(X = k)for the binomial distribution, B(n,p) wheren = 6,p = \(\frac{1}{3}\),k=3
The probability distribution of a random variable X is given below.

X

0

1

2

3

P(X)

K

\(\frac{k}{2}\)

\(\frac{k}{4}\)

\(\frac{k}{8}\)

i) Find k
ii) P(X > 2)

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

Define Probability Density function

An urn contains 5 mangoes and 4 apples Three fruits are taken at randaom If the number of apples
taken is a random variable, then find the values of the random variable and number of points in its
inverse images.

Prove that Var(x)=E(X²)-{E(X)]²

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

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

Part - B

Generic 5 - Answer All Question

Four bad oranges are accidentally mixed with sixteen good ones. Find the probability distribution of
bad oranges in a draw of two oranges. Also find the mean, variance and standard deviation of the
distribution.

A box contains 4 red and 5 black marbles. Find the probability distribution of the red marbles in a
random draw of three marbles. Also find the mean and standard deviation of the distribution

The p.d.f of a continuous random variable X is f(x) = where a and b are some
constants. Find

(a) a and b if E(x) = \(\frac{3}{5}\)

(b) Var (X)

Suppose that a pair of fair dice are tossed and let random variable X denote the sum of outcomes.
Find the mean and variance of the probability distribution of X.

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

The probability density function of random variable X is given by f(x) = \(\begin{cases} k \quad1 ≤ x ≤ 5\\0 \quad otherwise \end{cases}\)Find
(i) Distribution function
(ii) P(X < 3)
(iii) P(2 < X < 4)
(iv) P(3 ≤ X )

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.

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.

The probability density function random variable X is given by f(x) =\(\begin{cases} 16xe^{-4x} \quad for x > 0 \\0 \quad for x ≤ 0 \end{cases}\) find the
mean and variance of X.

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?

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X	0	1	2	3
P(X)	K	\(\frac{k}{2}\)	\(\frac{k}{4}\)	\(\frac{k}{8}\)

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12th Maths Monthly Test - 2 ( Probability Distributions)

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 2 ( Probability Distributions)-Aug 2020

Part - A

Generic 2 - Answer All Question

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.

Compute P(X = k)for the binomial distribution, B(n,p) where n=9,p =\(\frac{1}{2}\),k=7

Define discrete random variable

In an investment a man can make a profit of Rs.5000 with a probability of 0.62 or a loss of Rs.8000 with a probability of 0.38. Find the expected gain or loss %

Suppose X is the number of tails occurred when three fair coins are tossed once simultaneously. Find the values ofthe random variable X and number of points in its inverse images.

Define Distribution function.

Find the binomial distribution function for each of the following. (i) Five fair coins are tossed once and X denotes the number of heads. (ii) A fair die is rolled 10 times and X denotes the number of times 4 appeared.

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

Define Variance of a random variable X?

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.

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

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.

What is the Probability Mass function of a binomial random variable?

For the random variable X with the given probability mass function as below, find the mean and variance. \(f(x)=\begin{cases}\frac{4-x}{6}x=1,2,3 \end{cases}\)

A six sided die is marked '1' on one face, '3' on two of its faces, and '5' on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find (i) the probability mass function (ii) the cumulative distribution function (iii) P(4 ≤ X < 10) (iv) P(X ≥ 6)

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 .

Define a random variable

Compute P(X = k)for the binomial distribution, B(n,p) wheren = 6,p = \(\frac{1}{3}\),k=3

The probability distribution of a random variable X is given below. X 0 1 2 3 P(X) K \(\frac{k}{2}\) \(\frac{k}{4}\) \(\frac{k}{8}\) i) Find k ii) P(X > 2)

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

Define Probability Density function

An urn contains 5 mangoes and 4 apples Three fruits are taken at randaom If the number of apples taken is a random variable, then find the values of the random variable and number of points in its inverse images.

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

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

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

Part - B

Generic 5 - Answer All Question

Four bad oranges are accidentally mixed with sixteen good ones. Find the probability distribution of bad oranges in a draw of two oranges. Also find the mean, variance and standard deviation of the distribution.

A box contains 4 red and 5 black marbles. Find the probability distribution of the red marbles in a random draw of three marbles. Also find the mean and standard deviation of the distribution

The p.d.f of a continuous random variable X is f(x) = where a and b are some constants. Find (a) a and b if E(x) = \(\frac{3}{5}\) (b) Var (X)

Suppose that a pair of fair dice are tossed and let random variable X denote the sum of outcomes. Find the mean and variance of the probability distribution of X.

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

The probability density function of random variable X is given by f(x) = \(\begin{cases} k \quad1 ≤ x ≤ 5\\0 \quad otherwise \end{cases}\)Find (i) Distribution function (ii) P(X < 3) (iii) P(2 < X < 4) (iv) P(3 ≤ X )

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.

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.

The probability density function random variable X is given by f(x) =\(\begin{cases} 16xe^{-4x} \quad for x > 0 \\0 \quad for x ≤ 0 \end{cases}\) find the mean and variance of X.

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?

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Tamilnadu Stateboardszs - Class

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.

In an investment a man can make a profit of Rs.5000 with a probability of 0.62 or a loss of Rs.8000
with a probability of 0.38. Find the expected gain or loss %

Suppose X is the number of tails occurred when three fair coins are tossed once simultaneously.
Find the values ofthe random variable X and number of points in its inverse images.

Find the binomial distribution function for each of the following.
(i) Five fair coins are tossed once and X denotes the number of heads.
(ii) A fair die is rolled 10 times and X denotes the number of times 4 appeared.

A random variable X has the following probability mass function.

X

1

2

3

4

5

f(X)

K²

2k²

3k²

2k

3k

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.

For the random variable X with the given probability mass function as below, find the mean and
variance. \(f(x)=\begin{cases}\frac{4-x}{6}x=1,2,3 \end{cases}\)

A six sided die is marked '1' on one face, '3' on two of its faces, and '5' on remaining three faces. The
die is thrown twice. If X denotes the total score in two throws, find
(i) the probability mass function
(ii) the cumulative distribution function
(iii) P(4 ≤ X < 10)
(iv) P(X ≥ 6)

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 .

The probability distribution of a random variable X is given below.

X

0

1

2

3

P(X)

K

\(\frac{k}{2}\)

\(\frac{k}{4}\)

\(\frac{k}{8}\)

i) Find k
ii) P(X > 2)

An urn contains 5 mangoes and 4 apples Three fruits are taken at randaom If the number of apples
taken is a random variable, then find the values of the random variable and number of points in its
inverse images.

Prove that Var(x)=E(X²)-{E(X)]²

Four bad oranges are accidentally mixed with sixteen good ones. Find the probability distribution of
bad oranges in a draw of two oranges. Also find the mean, variance and standard deviation of the
distribution.

A box contains 4 red and 5 black marbles. Find the probability distribution of the red marbles in a
random draw of three marbles. Also find the mean and standard deviation of the distribution

The p.d.f of a continuous random variable X is f(x) = where a and b are some
constants. Find

(a) a and b if E(x) = \(\frac{3}{5}\)

(b) Var (X)

Suppose that a pair of fair dice are tossed and let random variable X denote the sum of outcomes.
Find the mean and variance of the probability distribution of X.

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

The probability density function of random variable X is given by f(x) = \(\begin{cases} k \quad1 ≤ x ≤ 5\\0 \quad otherwise \end{cases}\)Find
(i) Distribution function
(ii) P(X < 3)
(iii) P(2 < X < 4)
(iv) P(3 ≤ X )

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.

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.

The probability density function random variable X is given by f(x) =\(\begin{cases} 16xe^{-4x} \quad for x > 0 \\0 \quad for x ≤ 0 \end{cases}\) find the
mean and variance of X.

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?