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12th Maths Weekly Test-3 (Probability Distributions)

St. Britto Hr. Sec. School - Madurai

12th Maths Weekly Test-3 (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 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) = \(\begin {cases} \frac{2}{x^3}\space 0<x>l \\0 \space \space 1 ≤ x < 2l \end {cases}\)

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

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 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

1
2
3
4

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

6
4
3
2

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 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

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

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

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

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

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

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

If X is a binomial randam variable with expected value 6 and variance 2.4, then P(X=5) is

\(\left(\frac{10}{5}\right)^2\left(\frac{3}{5}\right)^6\left(\frac{2}{5}\right)^4\)
\(\left(\frac{10}{5}\right)\left(\frac{3}{5}\right)^5\)
\(\left(\frac{10}{5}\right)\left(\frac{3}{5}\right)^4\left(\frac{2}{5}\right)^6\)
\(\left(\frac{10}{5}\right)\left(\frac{3}{5}\right)^5\left(\frac{2}{5}\right)^5\)

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

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

1
2
3
4

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

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

1
2
3
4

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

X

-2

-1

0

1

2

f(X)

K

2k

3k

4k

5k

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

Let X have a Bernoulli distribution with mean 0.4, then the variance of (2X - 3) is

0.24
0.48
0.6
0.96

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

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

Define Probability Mass Function.

Define Probability Density function

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

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

Prove that Var(ax+b)=a² Var(X)

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

A coin is tossed twice. If X is a random variable defined as the number of heads minus the number of
tails, then obtain its probability distribution.

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

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 %

Part - C

Generic 3 - Answer All Question

Give any three properties on expectation and variance.

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.

Find the variance of the probability distribution

X

0

1

2

3

4

5

P(x)

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

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

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

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

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

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

Find the mean, variance and standard deviation of the number of heads in two tosses of a coin

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

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

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?

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.

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