12th Maths Weekly Test -1 (Probability Distributions)

St. Britto Hr. Sec. School - Madurai

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

Time : 1.30 Hours

Part - A

Multiple Choice Question - Answer All Question

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

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

If the function f(x) = \(\frac{1}{12}\)for a < x < b, represents a probability density function of a continuous random variable X, then which of the followingcannot be the value of a and b?

0 and 12
5 and 17
7 and 19
16 and 24

Two coins are to be flipped. The first coin will land on heads with probability 0.6, the second with probability 0.5. Assume that the results of the flips are independent, and let X equal the total number of heads that result The value of E[X] is
0.11

1.1

1.0

1
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
\(\frac{19}{6}\)

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

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

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

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

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

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

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

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

X

-2

-1

0

1

2

f(X)

K

2k

3k

4k

5k

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

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

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?

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

Part - B

Generic 2 - Answer All Question

Define a random variable

Define Distribution function.

Define Continuous random variable.

Define Variance of a random variable X?

Prove that Var(x)=E(X²)-{E(X)]²
Define discrete random variable

Part - C

Generic 3 - Answer All Question

Give any three properties of distribution function.

Two cards are drawn successively without replacement from a well shuffled pack of 52 cards. Find
the probability distribution of number of spades.

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?

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

A random variable X has the following probability distribution

X

0

1

2

3

4

5

6

7

P(x)

0

K

2K

2K

3K

K²

2K²

7K²+K

Evaluate (i) k

(ii) P(X ≥ 6)

(iii) P(0< X< 3)

Find the variance of the probability distribution

X

0

1

2

3

4

5

P(x)

\(\frac{1}{6}\)

\(\frac{5}{18}\)

\(\frac{2}{9}\)

\(\frac{1}{6}\)

\(\frac{1}{9}\)

\(\frac{1}{18}\)

Part - D

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

If X is a binomial random variable with mean 4 and variance 2 find P(|X − 2| ≤ 2)
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.

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X	0	1	2	3	4	5
P(x)	\(\frac{1}{6}\)	\(\frac{5}{18}\)	\(\frac{2}{9}\)	\(\frac{1}{6}\)	\(\frac{1}{9}\)	\(\frac{1}{18}\)

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12th Maths Weekly Test -1 (Probability Distributions)

St. Britto Hr. Sec. School - Madurai

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

Part - A

Multiple Choice Question - Answer All Question

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

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

If the function f(x) = \(\frac{1}{12}\)for a < x < b, represents a probability density function of a continuous random variable X, then which of the followingcannot be the value of a and b?

Two coins are to be flipped. The first coin will land on heads with probability 0.6, the second with probability 0.5. Assume that the results of the flips are independent, and let X equal the total number of heads that result The value of E[X] is

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

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

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.

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

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

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?

Part - B

Generic 2 - Answer All Question

Define a random variable

Define Distribution function.

Define Continuous random variable.

Define Variance of a random variable X?

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

Define discrete random variable

Part - C

Generic 3 - Answer All Question

Give any three properties of distribution function.

Two cards are drawn successively without replacement from a well shuffled pack of 52 cards. Find the probability distribution of number of spades.

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

A random variable X has the following probability distribution X 0 1 2 3 4 5 6 7 P(x) 0 K 2K 2K 3K K2 2K2 7K2+K Evaluate (i) k (ii) P(X ≥ 6) (iii) P(0< X< 3)

Find the variance of the probability distribution X 0 1 2 3 4 5 P(x) \(\frac{1}{6}\) \(\frac{5}{18}\) \(\frac{2}{9}\) \(\frac{1}{6}\) \(\frac{1}{9}\) \(\frac{1}{18}\)

Part - D

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

If X is a binomial random variable with mean 4 and variance 2 find P(|X − 2| ≤ 2)

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.

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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

Prove that Var(x)=E(X²)-{E(X)]²

Two cards are drawn successively without replacement from a well shuffled pack of 52 cards. Find
the probability distribution of number of spades.

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

A random variable X has the following probability distribution

X

0

1

2

3

4

5

6

7

P(x)

0

K

2K

2K

3K

K²

2K²

7K²+K

Evaluate (i) k

(ii) P(X ≥ 6)

(iii) P(0< X< 3)

Find the variance of the probability distribution

X

0

1

2

3

4

5

P(x)

\(\frac{1}{6}\)

\(\frac{5}{18}\)

\(\frac{2}{9}\)

\(\frac{1}{6}\)

\(\frac{1}{9}\)

\(\frac{1}{18}\)

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

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.