Exercise 6.3 - Chapter 6 Random Variable and Mathematical Expectation 12th Maths Guide Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated On May 15, 2024

By SaraNextGen

Ex $6.3$
Choose the correct answer:
Question 1.
The value which is obtained by multiplying possible values of a random variable with a probability of occurrence and is equal to the weighted average is called
(a) Discrete value
(b) Weighted value
(c) Expected value
(d) Cumulative value
Answer:
(c) Expected value
Question 2.
Demand of products per day for three days are $21,19,22$ units and their respective probabilities are $0.29,0.40,0.35$. Profit per unit is $0.50$ paisa then expected profits for three days are
(a) $21,19,22$
(b) $21.5,19.5,22.5$
(c) $0.29,0.40,0.35$
(d) $3.045,3.8,3.85$
Answer:
(d) $3.045,3.8,3.85$
Hint:
The expected profit for three days are as follows:
For day $1 \Rightarrow 21 \times 0.29 \times 0.5=3.045$
For day $2 \Rightarrow 19 \times 0.4 \times 0.5=3.8$
For day $3 \Rightarrow 22 \times 0.35 \times 0.5=3.85$
Question 3.
Probability which explains $\mathrm{x}$ is equal to or less than particular value is classified as
(a) discrete probability
(b) cumulative probability
(c) marginal probability
(d) continuous probability
Answer:

(b) cumulative probability
Question 4.
Given $\mathrm{E}(\mathrm{X})=5$ and $\mathrm{E}(\mathrm{Y})=-2$, then $\mathrm{E}(\mathrm{X}-\mathrm{Y})$ is
(a) 3
(b) 5
(c) 7
(d) $-2$
Answer:
(c) 7
Hint:
$
\mathrm{E}(\mathrm{X}-\mathrm{Y})=\mathrm{E}(\mathrm{X})-\mathrm{E}(\mathrm{Y})=5-(-2)=7 \text {. }
$
Question 5.
A variable that can assume any possible value between two points is called
(a) discrete random variable
(b) continuous random variable
(c) discrete sample space
(d) random variable
Answer:
(b) continuous random variable
Question 6.
A formula or equation used to represent the probability distribution of a continuous random variable is called
(a) probability distribution
(b) distribution function
(c) probability density function
(d) mathematical expectation
Answer:
(c) probability density function

Question 7.
If $\mathrm{X}$ is a discrete random variable and $\mathrm{p}(\mathrm{x})$ is the probability of $\mathrm{X}$, then the expected value of this random variable is equal to
(a) $\Sigma \mathrm{f}(\mathrm{x})$
(b) $\Sigma[x+f(x)]$
(c) $\sum f(x)+x$
(d) $\operatorname{\Sigma xP}(\mathrm{x})$
Answer:
(d) $\operatorname{\Sigma xP}(\mathrm{x})$
Question 8.
Which of the following is not possible in probability distribution?
(a) $\Sigma \mathrm{p}(\mathrm{x}) \geq 0$
(b) $\Sigma p(x)=1$
(c) $\Sigma \mathrm{xp}(\mathrm{x})=2$
(d) $\mathrm{p}(\mathrm{x})=-0.5$
Answer:
(d) $\mathrm{p}(\mathrm{x})=-0.5$
Hint:
$p(x)=-0.5$ is not possible since the probability cannot be negative.
Question 9.
If $\mathrm{c}$ is a constant, then $\mathrm{E}(\mathrm{c})$ is
(a) 0
(b) 1
(c) $\mathrm{c} \mathrm{f}(\mathrm{c})$
(d) $\mathrm{c}$
Answer:
(d) $\mathrm{c}$

Question 10.
A discrete probability distribution may be represented by
(a) table
(b) graph
(c) mathematical equation
(d) all of these
Answer:
(d) all of these
Question 11.
A probability density function may be represented by
(a) table
(b) graph
(c) mathematical equation
(d) both (b) and (c)
Answer:
(d) both (b) and (c)
Question 12.
If $\mathrm{c}$ is a constant in a continuous probability distribution, then $\mathrm{p}(\mathrm{x}=\mathrm{c})$ is always equal to
(a) zero
(b) one
(c) negative
(d) does not exist
Answer:
(a) zero

Question 13.
$\mathrm{E}[\mathrm{X}-\mathrm{E}(\mathrm{X})]$ is equal to
(a) $\mathrm{E}(\mathrm{X})$
(b) $\mathrm{V}[\mathrm{X}]$
(c) 0
(d) $\mathrm{E}(\mathrm{X})-\mathrm{X}$
Answer:
(c) 0
Hint:
$
\mathrm{E}[\mathrm{X}-\mathrm{E}(\mathrm{X})]=\mathrm{E}(\mathrm{X})-\mathrm{E}[\mathrm{E}(\mathrm{X})]=\mathrm{E}(\mathrm{X})-\mathrm{E}(\mathrm{X})=0
$
Question 14.
$\mathrm{E}[\mathrm{X}-\mathrm{E}(\mathrm{X})]^2$ is
(a) $\mathrm{E}(\mathrm{X})$
(b) $\mathrm{E}\left(\mathrm{X}^2\right)$
(c) $\mathrm{V}(\mathrm{X})$
(d) S.D (X)
Answer:
(c) $\mathrm{V}(\mathrm{X})$
Hint:
$
\begin{aligned}
& \mathrm{E}[\mathrm{X}-\mathrm{E}(\mathrm{X})]^2=\mathrm{E}\left[\mathrm{X}^2-2 \mathrm{XE}(\mathrm{X})+\mathrm{E}(\mathrm{X})^2\right] \\
& =\mathrm{E}\left[\mathrm{X}^2\right]-2[\mathrm{E}(\mathrm{X})]^2+[\mathrm{E}(\mathrm{X})]^2 \\
& =\mathrm{E}\left[\mathrm{X}^2\right]-[\mathrm{E}(\mathrm{X})]^2 \\
& =\operatorname{Var}(\mathrm{X})
\end{aligned}
$

Question 15 .
If the random variable takes negative values, then the negative values will have
(a) positive probabilities
(b) negative probabilities
(c) constant probabilities
(d) difficult to tell
Answer:
(a) positive probabilities
Question 16.
If we have $f(x)=2 x, 0 \leq x \leq 1$, then $f(x)$ is a
(a) probability distribution
(b) probability density function
(c) distribution function
(d) continuous random variable
Answer:
(b) probability density function
Question 17.
A discrete probability function $\mathrm{p}(\mathrm{x})$ is always
(a) non-negative
(b) negative
(c) one
(d) zero
Answer:
(a) non-negative
Question 18.
In a discrete probability distribution, the sum of all the probabilities is always equal to
(a) zero
(b) one
(c) minimum
(d) maximum
Answer:
(b) one

Question 19.
The expected value of a random variable is equal to its
(a) variance
(b) standard deviation
(c) mean
(d) covariance
Answer:
(c) mean
Question 20.
A discrete probability function $\mathrm{p}(\mathrm{x})$ is always non-negative and always lies between
(a) 0 and $\infty$
(b) 0 and 1
(c) $-1$ and $+1$
(d) $-\infty$ and $+\infty$
Answer:
(b) 0 and 1
Question 21.
The probability density function $\mathrm{p}(\mathrm{x})$ cannot exceed
(a) zero
(b) one
(c) mean
(d) infinity
Answer:
(b) one
Question 22.
The height of persons in a country is a random variable of the type
(a) discrete random variable
(b) continuous random variable
(c) both (a) and (b)
(d) neither (a) nor (b)
Answer:
(b) continuous random variable

Question 23.
The distribution function $\mathrm{F}(\mathrm{x})$ is equal to
(a) $\mathrm{P}(\mathrm{X}=\mathrm{x})$
(b) $P(X \leq x)$
(c) $\mathrm{P}(\mathrm{X} \geq \mathrm{x})$
(d) all of these
Answer:
(b) $\mathrm{P}(\mathrm{X} \leq \mathrm{x})$