Question 11 Mark
State the relation between mean and variance of binomial distribution.
Answer
In binomial distribution value of mean is always greater than its variance.
Hence, $np > npq$ OR $npq < np.$
View full question & answer→Question 21 Mark
State the relation between the probability of success and failure in Bernoulli trials.
AnswerIn Bernoulli trials success and failure are mutually exclusive and exhaustive events. Therefore $p + q = 1$ and $q = 1 - p.$
View full question & answer→Question 31 Mark
The parameters of a binomial distribution are $10$ and Calculate its variance.
Answer$
\begin{aligned}
& n=10, p=\frac{2}{5} \\
& \therefore q=1-p=1-\frac{2}{5}=\frac{3}{5} \\
& \text { Variance }=n p q \\
& =10 \times \frac{2}{5} \times \frac{3}{5}=\frac{12}{5}
\end{aligned}
$
Hence, the variance obtained is $\frac{12}{5}$.
View full question & answer→Question 41 Mark
Mean of a symmetrical binomial distribution is $7.$ Find the value of its parameter $n.$
AnswerHere, it is given that binomial distribution is symmetrical.
$\therefore p=q=\frac{1}{2}$
$ \text { Mean }=n p=7 \text { (Putting } p=\frac{1}{2} \text { ) }$
$ \therefore \mathrm{n} \times \frac{1}{2}=7$
$ \therefore \mathrm{n}=14$
Hence, the value of parameter $n$ obtained is $14 .$
View full question & answer→Question 51 Mark
State the formula to find variance of discrete variable.
AnswerThe formula to find the variance of discrete variable is as follows :
$\sigma 2 = V(X) = E(X2) – [E(X)]2$
$= \sum x^2p (x) – [\sum x ∙ p(x)]2$
View full question & answer→Question 61 Mark
State the formula to find Mean of discrete variable.
AnswerFormula to find Mean of discrete variable is as follows :
Mean $= \mu = E(X) =\sum \mathrm{x}_{\mathrm{i}} \mathrm{P}\left(\mathrm{x}_{\mathrm{i}}\right)$
This value is also called expected value of discrete variable $X$
View full question & answer→Question 71 Mark
Define discrete probability distribution.
Answer
Suppose, probability of a discrete random variable $X$ is $p\left[X=x_i\right]=p\left(x_i\right)$. If for $i=1,2,3$, $n ; p\left(x_i\right)>0$ and $\sum_{i=1}^n p\left(x_i\right)=1$,
then set of real values $\left\{p\left(x_1\right), p\left(x_2\right), \ldots \ldots ., p\left(x_n\right)\right\}$ is called probability distribution of a discrete random variable $X$. In tabular form it is written as under:
$\begin{array}{c|c|c|c|c|c|c|c}
\mathrm{X}=x_i & x_1 & x_2 & x_3 & \cdots \cdots \cdots \cdot & x_{n-1} & x_n & \text { Total} \\
\hline p\left(x_i\right) & p\left(x_1\right) & p\left(x_2\right) & p\left(x_3\right) & \cdots \cdots \cdots \cdot & p\left(x_{n-1}\right) & p\left(x_n\right) & \sum_i^n p\left(x_i\right)=1
\end{array}$
Here, $0 < p (x_i) < 1$ and $i = 1, 2, 3, …, n$
View full question & answer→Question 81 Mark
Define continuous random variable.
AnswerA random variable $X$ which is capable of assuming any value of the interval.
View full question & answer→Question 91 Mark
The probability of failure in a binomial distribution is $0.6$ and the number of trials in it is $5.$ Find the probability of success.
AnswerThe relation between mean and variance of binomial distribution is $p + q = 1.$
Hence, $p + 0.6 = 1$
$\therefore p = 1 - 0.6 = 0.4$
The probability of success is $0.4.$
View full question & answer→Question 101 Mark
Define discrete random variable.
AnswerA random variable $X,$ Which is capable of taking any real value of set $R$ or its subset is called a discrete random variable.
View full question & answer→Question 111 Mark
The number of Bernoulli trials is $10$ in binomial probability distribution, state the value of interval of the number of successes $x$.
View full question & answer→Question 121 Mark
For a binomial probability distribution mean $=4$ and variance $=0.8$, find the value of $n$.
View full question & answer→Question 131 Mark
The parameters of binomial probability distribution $n=5$ and $q=\frac{1}{3}$, write the binomial probability distribution.
Answer$\overline{p(x)={ }^{5} c_{x}\left(\frac{2}{3}\right)^{x}\left(\frac{1}{3}\right)^{5-x}}$
View full question & answer→Question 141 Mark
State the probability distribution for Bernoulli trials.
AnswerThe probability distribution for Bernoulli trial is 'binomial probability distributioti'.
View full question & answer→Question 151 Mark
Random variable $X$ assume the values $0,1$ and $2$ with respective probability $k, \frac{k}{2}$ and $\frac{k}{3}$ find the value of $k$.
Answer$\frac{\overline{6}}{11}$
View full question & answer→Question 161 Mark
Random variable $X$ assume the values $1,2$ and $3$ with respective probability $0.25,0.40$ and $0.35$, find mean of $X$.
View full question & answer→Question 171 Mark
What Is the value of probability of success $p$ In the positively skewed binomial probability distribution?
AnswerThe value of probability of success $p<0.5$, in the positively skewed binomial probability distribution.
View full question & answer→Question 181 Mark
In binomial distribution $q-2 p=\frac{3}{5} .$ If mean is $0.5$, find $n$.
View full question & answer→Question 191 Mark
If in a binomial probability distribution $n =8$ and $p-q=\frac{1}{4}$ find the probability of success.
View full question & answer→Question 201 Mark
If in a binomial probability distribution $n=6$ and $p=0.6$, then find its standard deviation.
View full question & answer→Question 211 Mark
How are the probability of success and failure always In binomial probability distribution?
AnswerIn binomial probability distribution, always the probability of success and failure are not equal.
View full question & answer→Question 221 Mark
In a binomial distribution $n=10, q=\frac{3}{4}$ Find mean of the distribution.
View full question & answer→Question 231 Mark
"In a binomial distribution its mean is $10$ and standard deviation is $5.$" Examine the validity of this statement.
AnswerHere, mean $(n p)=10$ and $S . d . \sqrt{n p q}=5$
View full question & answer→Question 241 Mark
In binomial probability distribution if $p=\frac{1}{3}$, then what will you say about its skewness?
Answer$p=\frac{1}{3}$, is less than $\frac{1}{2} .$ Hence there is positive skewness in binomial probability distribution.
View full question & answer→Question 251 Mark
When will the binomial probability distribution become a symmetrical distribution?
AnswerIn binomial probability distribution when $p=1 / 2$, then it will becomes a symmetrical distribution.
View full question & answer→Question 261 Mark
Represent mathematically the relation between variance and mean of binomial probability distribution.
AnswerIn binomial probability distribution, variance $(n p q)$
View full question & answer→Question 271 Mark
State the mean and variance of binomial probability distribution.
AnswerThe mean and variance of binomial probability distribution are , $np$ and $npq$ respectively.
View full question & answer→Question 281 Mark
State the standard deviation of binomial probability distribution.
AnswerThe standard deviation of binomial probability distribution is $\sqrt{n p q}$
View full question & answer→Question 291 Mark
Write the probability function of binomial probability distribution.
Answer$P ( X = x )= p ( x )={ }^{n} c_{x} p^{x} q^{n-x}$
View full question & answer→Question 301 Mark
What is a binomial random variable?
AnswerIn n Bernoulli trials if the number of success is denoted by $X$, then $X$ is called a binomial random variable.
View full question & answer→Question 311 Mark
The probability of failure $P(F)=q$. What will you say about the value of $q?$
AnswerIf the probability of failure $P(F)=q$, then $0$
View full question & answer→Question 321 Mark
State the minimum value of the probability of success $p$.
AnswerThe minimum value of the probability of success $p$ is $0 .$
View full question & answer→Question 331 Mark
In a dichotomous experiment if the probability of success at any trial out of $n$ trials remains constant, then what are these $n$ trials called?
AnswerIn a dichotomous experiment if the probability of success at any trial out of $n$ trials remains constant, then these $n$ trials are called Bernoulli trials.
View full question & answer→Question 341 Mark
If $P(S)=p$ and $P(F)=q$, then what is the value of $p+q$ ?
AnswerIf $P(S)=p$ and $P(F)=q$, then $p+q=1$.
View full question & answer→Question 351 Mark
How are results of a dichotomous experiments?
AnswerThe results $S$ and $F$ of a dichotomous experiment are mutually exclusive.
View full question & answer→Question 361 Mark
Write the sample space corresponding to a dichotomous experiment.
AnswerThe sample space corresponding to a dichotomous experiment is $U=\{S, F\} ;$ where $S=$ Success, $F=$ Failure.
View full question & answer→Question 371 Mark
What are success and failure?
AnswerA definite outcome which occurs in a random experiment is called success and If it does not occur then it is called failure.
View full question & answer→Question 381 Mark
What is a dichotomous experiment?
AnswerAn experiment with two outcomes as success and failure is called dichotomous experiment.
View full question & answer→Question 391 Mark
For random variable $X, E(X)=1.2$ and variance $=0.6$, find $E\left(X^{2}\right)$
View full question & answer→Question 401 Mark
For a random variable $X=\{-1,0,1\}$, if $p(1)=\frac{1}{3}, p(0)=\frac{1}{6}$ and $p(-1)=\frac{2}{3} .$ State whether $p(x)$ is a probability function of $x$ or not.
Answer$\overline{p(x)}$ is not probability function.
View full question & answer→Question 411 Mark
How is always the value of the variance of the distribution of a random variable $X$ ?
AnswerThe value of the variance of the distribution of a random variable $X$ is always positive.
View full question & answer→Question 421 Mark
State the formula of $E(X)$ and $E\left(X^{2}\right)$ for random variable $X$.
AnswerFormula for random variable $X$ is $E(X)=\sum x . p(x)$ and $E\left(X^{2}\right)=\sum x^{2} p(x)$
View full question & answer→Question 431 Mark
What is the symbol for the mean and variance of discrete random variable?
AnswerThe symbol for the mean is denoted by $\mu$ and variance of discrete random variable by $\sigma^{2}$.
View full question & answer→Question 441 Mark
What is expected value of discrete random variable $X$ called?
AnswerExpected value of discrete random variable $X$ is called the mean of discrete random variable of $X$.
View full question & answer→Question 451 Mark
State the conditions that $p(x)$ becomes the probability distribution.
AnswerIf for a random variable $X(1)$ for each value of $X p(x)>0$ and $(2) \sum p(x)=1$, then $p(x)$ Is the probability distribution of the random variable $X$.
View full question & answer→Question 461 Mark
What is the symbol for the probability distribution of a random variable $X?$
AnswerThe symbol for the probability distribution of a random variable $X$ is $p(x)$.
View full question & answer→Question 471 Mark
What is a continuous random variable?
AnswerA random variable $X$ which assumes any value in the interval $(a, b), a$
View full question & answer→Question 481 Mark
Give two illustrations of a discrete random variable.
AnswerTwo illustrations of a discrete random variables are: $(1)$ The number of printing mistakes in a book and $(2)$ The number obtained on upper side of die in the experiment of throwing a die.
View full question & answer→Question 491 Mark
Denote symbolically random variable $X$ defined on sample space $U$.
AnswerA random variable $X$ defined on sample space $U$ is denoted symbolically as $X: U \rightarrow R$.
View full question & answer→Question 501 Mark
What is a random variable?
AnswerThe function $X$ which transforms each primary outcome of a sample space $U$ into real number Is called a random variable.
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