Solve the Following Question.(3 Marks)

Question 13 Marks

$P(X=x)= \begin{cases}\left(\begin{array}{l}5 \\ x\end{array}\right) \frac{1}{2^5}, & x=0,1,2,3,4,5 \\ 0 & \text { otherwise }\end{cases}$ Show that $P(X \leq 2) = P(X \geq 3).$

View full question & answer→

Question 23 Marks

An examination consists of 5 multiple choice questions, in each of which the candidate has to decide which one of 4 suggested answers is correct. A completely unprepared student guesses each answer randomly. Find the probability that this student gets 4 or more correct answers.

View full question & answer→

Question 33 Marks

If $X$ has Poisson distribution with parameter $m$, such that
$\frac{P(X=x+1)}{P(X=x)}=\frac{m}{x+1}$
find probabilities $P(X=1)$ and $P(X=2)$, when $X$ follows Poisson distribution with $m =2$ and $P ( X =0)=0.1353$.

View full question & answer→

Question 43 Marks

It is observed that it rains on 10 days out of 30 days. Find the probability that
(i) it rains on exactly 3 days of a week.
(ii) it rains at most 2 days a week.

View full question & answer→

Question 53 Marks

In a large school, 80% of the students like mathematics. A visitor asks each of 4 students, selected at random, whether they like mathematics, (i) Calculate the probabilities of obtaining an answer yes from all of the selected students, (ii) Find the probability that the visitor obtains the answer yes from at least 3 students.

View full question & answer→

Question 63 Marks

A computer installation has 3 terminals. The probability that anyone terminal requires attention during a week is 0.1, independent of other terminals. Find the probabilities that (i) 0 (ii) 1 terminal requires attention during a week.

View full question & answer→

Question 73 Marks

Let X ~ B(10, 0.2). Find (i) P(X = 1) (ii) P(X ≥ 1) (iii) P(X ≤ 8)

View full question & answer→

Question 83 Marks

The probability distribution of a discrete r.v. X is as follows.

X	1	2	3	4	5	6
P(X=x)	K	2K	3K	4K	5K	6K

(i) Determine the value of k.
(ii) Find P(X ≤ 4), P(2 < X < 4), P(X ≥ 3).

View full question & answer→

Question 93 Marks

The $p.d.f.$ of the $r.v. x.$ is given by $ f_x(x)=\left\{\begin{array}{l} \frac{k}{\sqrt{x}}, 00, \text { otherwise } \end{array}\right. $ Determine $k, c.d.f.$ of $X$ and hence find $P(X \leq 2)$ and $P(X \geq 1).$

Answer

We know that
$\int_0^4 \frac{k}{\sqrt{x}} d x=1$
$\therefore k \int_0^4 x^{\frac{-1}{2}} d x=1$
$\therefore k\left[\frac{x^{\frac{1}{2}}}{\frac{1}{2}}\right]_0^4=1$
$\therefore 2 k[\sqrt{4}-\sqrt{0}] =1$
$\therefore 4 k =1$
$\therefore k =\frac{1}{4}$
$\text{c.d.f.}$ of $f(x)$ is
$F(x)=\int_0^x \frac{k}{\sqrt{x}} d x$
$\therefore F(x) =\int_0^x \frac{1}{4} x^{\frac{-1}{2}} d x=\frac{1}{4}\left[\frac{x^{\frac{1}{2}}}{\frac{1}{2}}\right]_0^x=\frac{\sqrt{x}}{2}$
$P(X \leq 2) =F(2)$
$ =\frac{\sqrt{2}}{2}=0.7071$
$P(X \geq 1) =1-P(X<1)=1-F(1)$
$ =1-\frac{1}{2}$
$=\frac{1}{2}$
$=0.5$

View full question & answer→

Question 103 Marks

Suppose error involved in making a certain measurement is a continuous r.v. X with p.d.f.
$
f(x)= \begin{cases}k\left(4-x^2\right) & \text { for }-2 \leq x \leq 2 \\ 0 & \text { otherwise }\end{cases}
$

Compute (i) $P ( X >0$ ), (ii) $P (-1< X <1$ ), (iii) $P ( X <-0.5$ or $X >0.5)$

View full question & answer→

Question 113 Marks

A player tosses two coins. He wins ₹ 10 if 2 heads appear, ₹ 5 if 1 head appears, and ₹ 2 if no head appears. Find the expected value and variance of the winning amount.

View full question & answer→

Question 123 Marks

Find the expected value and variance of the r.v. X if its probability distribution is as follows.

X	0	1	2	3	4	5
P(X=x)	1/32	5/32	10/32	10/32	5/32	1/32

View full question & answer→

Question 133 Marks

Find the expected value and variance of the r.v. X if its probability distribution is as follows.

X	1	2	3	...	n
P(X=x)	1/n	1/n	1/n	...	1/n

View full question & answer→

Question 143 Marks

It is known that, in a certain area of a large city, the average number of rats per bungalow is five. Assuming that the number of rats follows Poisson distribution, find the probability that a randomly selected bungalow has
(i) exactly $5$ rats
(ii) more than $5$ rats
(iii) between $5$ and $7$ rats, inclusive. Given $e^{-5} = 0.0067.$

View full question & answer→

Question 153 Marks

In a multiple-choice test with three possible answers for each of the five questions, what is the probability of a candidate getting four or more correct answers by random choice?

View full question & answer→

Question 163 Marks

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Find the probability that (i) all the five cards are spades, (ii) only 3 cards are spades, (iii) none is a spade.

View full question & answer→

Question 173 Marks

A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of (i) 2 successes (ii) at least 3 successes (iii) at most 2 successes.

View full question & answer→

Question 183 Marks

Following is the p.d.f. of a continuous r.v. X.
$
f(x)= \begin{cases}\frac{x}{8} & \text { for } 0<x<4 \\ 0 & \text { otherwise }\end{cases}
$
(i) Find an expression for the c.d.f. of $X$.
(ii) Find $F(x)$ at $x=0.5,1.7$, and 5 .

View full question & answer→

Question 193 Marks

Let X be the amount of time for which a book is taken out of the library by a randomly selected student and suppose that X has p.d.f.

$f(x)= \begin{cases}0.5 x & \text { for } 0 \leq x \leq 2 \\ 0 & \text { otherwise }\end{cases}$

Calculate (i) P(X ≤ 1), (ii) P(0.5 ≤ X ≤ 1.5), (iii) P(X ≥ 1.5).

View full question & answer→

Question 203 Marks

Find $k$, if the following function represents the p.d.f. of a r.v. X.
(i) $f(x)= \begin{cases}k x & \text { for } 0<x<2 \\ 0 & \text { otherwise }\end{cases}$
Also find $P\left[\frac{1}{4}<X<\frac{1}{2}\right]$

View full question & answer→

Question 213 Marks

It is felt that error in measurement of reaction temperature (in Celsius) in an experiment is a continuous r.v. with p.d.f.
$
f(x)= \begin{cases}\frac{x^3}{64} & \text { for } 0 \leq x \leq 4 \\ 0 & \text { otherwise }\end{cases}
$
(i) Verify whether $f(x)$ is a p.d.f.
(ii) Find $P (0< X \leq 1)$.
(iii) Find the probability that $X$ is between 1 and 3 .

View full question & answer→

Question 223 Marks

The following is the p.d.f. of a r.v. $X$. $ f(x)=\left\{\begin{array}{ll} \frac{x}{8} & \text { for } 0<x<4 \\ 0 & \text { otherwise } \end{array}\right. $ Find (i) $P ( X <1.5$ ), (ii) $P (1< X <2$ ), (iii) $P ( X >2)$

View full question & answer→

Question 233 Marks

Check whether each of the following is p.d.f.
f(x) = 2 for 0 < x < 1

Answer

The given function is $ \begin{aligned} & f ( x )=2 \text { for } 0< x <1 \text { Each } f ( x )>0, \\ & \text { but } \int_0^1 f(x) d x=\int_0^1 2 d x=2[x]_0^1 \\ &=2(1)=2>1 . \end{aligned} $ $\therefore$ The given function is not a p.d.f.

View full question & answer→

Question 243 Marks

Check whether each of the following is $f(x)= \begin{cases}x & \text { for } 0 \leq x \leq 1 \\ 2-x & \text { for } 1<x \leq 2\end{cases}$

Answer

Given function is
$f ( x )= x _1 0 \leq x \leq 1$
Each $f(x) \geq 0$, as $x \geq 0$.
and $\begin{aligned} \int_0^1 f(x) d x & =\int_0^1 x d x \\ &\end{aligned} $
$=\left[\frac{x^2}{2}\right]_0^1 $
$=\frac{1}{2} $
Also, $f(x)=2-x, 1 \leq x \leq 2$
$\Rightarrow$ Each $f(x) \geq 0,$ as $x \leq 2.$
and $\int_1^2 f(x) d x =\int_1^2(2-x) d x$
$ =2 \int_1^2 1 d x-\int_1^2 x d x $
$ =2[x]_1^2-\left(\frac{x^2}{2}\right)_1^2$
$=2(2-1)-\frac{1}{2}(4-1)$
$=1-\frac{3}{2}$
$=\frac{1}{2} $
Now, for the total range of $0 \leq x \leq 2$.
$ \int_0^2 f(x) d x =\int_0^1 f(x) d x+\int_1^2 f(x) d x$
$ =\frac{1}{2}+\frac{1}{2}$
$ =1 \text { } $
$\therefore$ The given function is a $p.d.f.$ of $x$.

View full question & answer→

Question 253 Marks

If a r.v. X has p.d.f. $ f(x)=\left\{\begin{array}{ll} \frac{c}{x} & \text { for } 1<x<3, c>0 \\ 0 & \text { otherwise } \end{array}\right. $ Find $c, E(X)$ and $V(X)$. Also find $f(x)$.

View full question & answer→

Question 263 Marks

Find the probability distribution of
(i) number of heads in two tosses of a coin,
(ii) number of trails in three tosses of a coin,
(iii) number of heads in four tosses of a coin.

View full question & answer→

Question 273 Marks

Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers. Find E(X).

View full question & answer→

Question 283 Marks

Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.

View full question & answer→

Question 293 Marks

Find expected value and variance of X, the number on the uppermost face of a fair die.

View full question & answer→

Maths (commerce) Solve the Following Question.(3 Marks)

Test yourself on this topic