CBSE Questions for Class 12 Commerce Applied Mathematics Probability Distribution And Its Mean And Variance Quiz 1 - MCQExams.com

The p.m.f.of a r.v. X is as follows ;
$$P(X=0)=3{ K }^{ 3 }\quad ,\quad P(X=1)=4K-10{ K }^{ 2 },\quad P(X=2)=5K-1,\quad P(X-x)=0$$ for any other value of x. then k is equal to 
  • 1
  • 2
  • $$\frac { 1 }{ 3 } $$
  • $$\frac { 2 }{ 3 } $$
Variance of the random variable $$X$$ is $$4$$. Its mean is $$2$$. Then $$E(X^{2})$$ is:
  • $$6$$
  • $$8$$
  • $$2$$
  • $$4$$
Another name for the mean of a probability distribution is expected value.
  • True
  • False
The variance of the random variable $$x$$ whose probability distribution is given by
$$X=x_{i}: \quad-1 \quad, 0, \quad  +1$$
$$p(X=x_{i}): 0.4,  \ 0.2,  \ \ \ 0.4 $$ is
  • $$0.4$$
  • $$0.6$$
  • $$0.8$$
  • $$1.0$$
To define probability disribution function we assign to each variable  
  • the respective probabilities
  • the specific random values
  • some integers
  • none
Out of following which are random variables
  • $$x=$$ "Number of heads when two coins are tossed.
  • $$x=$$"Sum of digits on uppermost face of two dice"
  • solution of "$$ X-4=0"$$
  • $$x=$$ "Raining"
Given $$E(X + c) = 8$$ and $$E(X - c) = 12$$ then the value of $$c$$ is
  • $$-2$$
  • $$4$$
  • $$-4$$
  • $$2$$
Which of the following is an example of a random experiment?
  • Selecting a card from a pack of playing cards.
  • Measuring the weight of a person.
  • Finding the length of your pencil box.
  • Throwing two coins together.
To verify Pythagoras theorem is a random experiment.
  • True
  • False
The mathematical expectation of sum of points when we throw n symmetrical dice is
  • 7n
  • $$7\displaystyle \times\frac{n}{2}$$
  • $$\displaystyle \frac{n}{2}$$
  • $$\displaystyle \frac{7n}{3}$$
Difference between sample space and subset of sample space is considered as
  • numerical complementary events
  • equal compulsory events
  • complementary events
  • compulsory events
In the p.d.f. of a random variable of three missing entries are in the ratio $$1:2:3$$.
$$X=x$$$$1$$$$2$$$$3$$$$4$$$$5$$
$$P(X=x)$$$$\dfrac{1}{10}$$---$$\dfrac{3}{20}$$
Then the missing entries are?
  • $$\dfrac{1}{5}, \dfrac{2}{5}, \dfrac{3}{5}$$
  • $$\dfrac{3}{12}, \dfrac{4}{15}, \dfrac{6}{15}$$
  • $$\dfrac{1}{10}, \dfrac{2}{10}, \dfrac{3}{10}$$
  • $$\dfrac{1}{8}, \dfrac{2}{8}, \dfrac{3}{8}$$
A random variable $$X$$ has the probability distribution $$X$$: $$1, 2, 3, 4, 5, 6, 7, 8 $$
$$P(X): 0.15, 0.23, 0.12, 0.10 ,0.20 ,0.08 ,0.07 ,0.05$$ . For the events $$\mathrm{E}=$$ $$\{$$X is a prime number $$\}$$ and $$\mathrm{F}=\{\mathrm{X}<4\}$$, the probability $$\mathrm{P}(\mathrm{E}\cup\mathrm{F})$$ is: 
  • $$0.87$$
  • $$0.77$$
  • $$0.35$$
  • $$0.50$$
Statement 1: The variance of first $$\mathrm{n}$$ even natural numbers is $$\displaystyle \frac{\mathrm{n}^{2}-1}{4}$$
Statement 2: The sum of first $$\mathrm{n}$$ natural numbers is $$\displaystyle \frac{\mathrm{n}(\mathrm{n}+1)}{2}$$ and the sum of squares of first $$\mathrm{n}$$ natural numbers is $$\displaystyle \frac{\mathrm{n}(\mathrm{n}+1)(2\mathrm{n}+1)}{6}$$
  • Statement 1 is true, Statement2 is true,Statement 2 is a correct explanation for Statement 1
  • Statement 1 is true, Statement2 is true;Statement2 is not a correct explanation for statement 1
  • Statement 1 is true, Statement 2 is false.
  • Statement 1 is false, Statement 2 is true
A random variable X has its range $$X = \{3, 2, 1\}$$ with the probabilities,
$$\dfrac{1}{2},\dfrac{1}{3}$$ and $$\dfrac{1}{6}$$ respectively. The mean value of X is
  • $$\dfrac{5}{3}$$
  • $$\dfrac{7}{3}$$
  • $$3$$
  • $$4$$
Four different objects 1,2,3,4 are distributed at random in four places marked 1,2,3,What is the probability that none of the objects occupy the place corresponding to its number ?
  • $$\frac{17}{24}$$
  • $$\frac{3}{8}$$
  • $$\frac{1}{2}$$
  • $$\frac{5}{8}$$
If the range of the random variable X is from a to $$\mathrm{b},\ \mathrm{a}<\mathrm{b},\ \mathrm{F}(\mathrm{X}<\mathrm{a})=$$
  • $$0$$
  • $$1$$
  • $$0.5$$
  • $$3$$
The cumulative distribution function of a random variable $$x$$ is defined as
  • $$P(X < x)$$
  • $$P(X \ge x)$$
  • $$P(X \le x)$$
  • $$P(X > x)$$
An urn contains $$5$$ red and $$2$$ black balls. Two balls are randomly drawn. Let $$X$$ represent the number of black balls. What are the possible values of $$X$$ ? Is $$X$$ a random variable ?
  • $$0,1,2$$
  • $$3,5,7$$
  • $$7,7,8$$
  • $$1,5,7$$
If an unbiased coin is tossed once, then the two events of getting a Head and a Tail are -
  • Mutually exclusive
  • Exhaustive
  • Equally likely
  • All of these
If a random variable X takes value $$ 0 $$ and $$1$$ with respective probabilities $$\dfrac{2}{3}$$ and $$\dfrac{1}{3}$$ , then the expected value of X is
  • $$\dfrac{2}{3}$$
  • $$\dfrac{1}{3}$$
  • $$0$$
  • $$1$$
In a class of 80 students, 48 are boys and the rest of the students are girls. If 10 students shift to the other class room and a student is selected at random from the remaining class, what is the probability that a girl is selected ?
  • 11/35
  • 27/70
  • 11/40
  • Cannot be determined

10 different books and 2 different pens are given to 3 boys so that each gets equal number of things. The probability that the same boy does not receive both the pens is
  • $$\frac{5}{11}$$
  • $$\frac{7}{11}$$
  • $$\frac{2}{3}$$
  • $$\frac{8}{11}$$
If the range of the random variable X is from $$a$$ to $$b$$ then $$F(X\leq b)$$ is
  • 0
  • 1
  • 0.5
  • 3
Expected number of heads when we toss $$n$$ unbiased coins is
  • $$2n$$
  • $$n$$
  • $$\dfrac{n}{2}$$
  • $$\dfrac{n}{4}$$
In $$5$$ throws of a die, getting $$1$$ or $$2$$ is a success. The mean number of successes is
  • $$\dfrac{5}{3}$$
  • $$\dfrac{3}{5}$$
  • $$\dfrac{5}{9}$$
  • $$\dfrac{9}{5}$$
Which of the following is NOT a random experiment ?
  • Rolling an unbiased dice
  • Tossing a fair coin
  • Drawing a card from a well shuffled pack of 52 card
  • None of these
The probability distribution of a discrete random variable $$X$$ is:
$$X = x$$$$1$$$$2$$$$3$$$$4$$$$5$$
$$P(X = x)$$$$k$$$$2k$$$$3k$$$$4k$$$$5k$$
Find $$P (X\leq 4)$$
  • $$\cfrac 23$$
  • $$\cfrac 34$$
  • $$\cfrac 45$$
  • $$\cfrac 56$$
Which of the following is not a random experiment?
  • Tossing a coin
  • Rolling a dice
  • Choosing a card from a deck of 52 cards
  • Throw a stone from a roof of a building
Which of the following is an random experiment?
  • Rolling a pair of dice
  • Choosing $$2$$ marbles from a jar
  • Choosing a number at random from $$1$$ to $$10$$
  • Tossing two coins
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Practice Class 12 Commerce Applied Mathematics Quiz Questions and Answers