Probability Distributions - Maharashtra Board STD 12 Science Maths Questions

Q 1MCQ2 Marks

If $X$ is a random variable with probability mass function
$P (x)=k x \quad, \quad$ for $x=1,2,3$
$=0$. otherwise then $k=\ldots$

A
$\frac{1}{5}$
B
$\frac{1}{4}$
✓
$\frac{1}{6}$
D
$\frac{2}{3}$

Answer: C.

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Q 2MCQ2 Marks

If the $\text{p.m.f.}$ of a $\text{r.v. x}$ is
$P (x) =\frac{c}{x^3},$ for $x=1,2,3=0,$ otherwise,
then $E ( X )=.......$

A
$\frac{216}{251}$
✓
$\frac{294}{251}$
C
$\frac{297}{294}$
D
$\frac{294}{297}$

Answer: B.

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Q 3MCQ2 Marks

Let the $\text{p.m.f.}$ of a random variable $X$ be $-$
$P(x)=\frac{3-x}{10}, $ for $ x=-1,0,1,2$
$=0 $ otherwise Then $ E(X)$ is $..... $

A
$1$
B
$2$
✓
$0$
D
$-1$

Answer: C.

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Q 4MCQ2 Marks

A random variable $X$ has the following probability distribution :

$X =x$	-2	-1	0	1	2	3
$P (x)$	0.1	0.1	0.2	0.2	0.3	0.1

Then $E (x)=$

A
0.8
B
0.9
C
0.7
D
1.1

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Q 5MCQ2 Marks

The integrating factor of linear differential equation $\frac{d y}{d x}+y \sec x=\tan x$ is

A
$\sec x-\tan x$
B
$\sec x \cdot \tan x$
C
$\sec x+\tan x$
D
$\sec x \cdot \cot x$

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Q 6Solve the Following Question.(2 Marks)2 Marks

Find the probability distribution of number of heads in two tosses of a coin.

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Q 7Solve the Following Question.(2 Marks)2 Marks

Find $k$, such that the function
$
P (x)=\left\{\begin{array}{ll}
k\left(\frac{4}{x}\right) & ; \quad x=0,1,2,3,4, k>0 \\
0 ; & \text { otherwise, }
\end{array}\right.
$
is a probability mass function (p.m.f.)

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Q 8Solve the Following Question.(2 Marks)2 Marks

A random variable $X \sim N(0,1)$. Find $P(X>0)$ and $P(X<0)$

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Q 9Solve the Following Question.(2 Marks)2 Marks

Find expected value of the random variable $X$ whose probability mass function is :

$X =x$	1	2	3
$P ( X =x)$	$\frac{1}{5}$	$\frac{2}{5}$	$\frac{2}{5}$

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Q 10Solve the Following Question.(2 Marks)2 Marks

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 $( X \leq 4$ )

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Q 11Solve the Following Question.(3 Marks)3 Marks

The probability distribution of discrete random variable $X$ is as follows:

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

Find (1) the value of $k$
(ii) $P (2< X <4)$
(iii) $P ( X \geq 3$ )

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Q 12Solve the Following Question.(3 Marks)3 Marks

The probability distribution of $X$, the number of defects per $10$ metres of a fabric is given by

$x$	$0$	$1$	$2$	$3$	$4$
$P ( X =x)$	$0.45$	$0.35$	$0.15$	$0.03$	$0.02$

Find the varlance of $X$.

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Q 13Solve the Following Question.(3 Marks)3 Marks

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

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Q 14Solve the Following Question.(3 Marks)3 Marks

If $f(x)=k x \quad$, for $0<x<2$
$=0$, otherwise,
is the probability density function of a random variable $X$. then find :
(1) Value of $k$,
(ii) $P (1< X <2)$

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Q 15Solve the Following Question.(3 Marks)3 Marks

For the following probability density function (p. d. f) of X, find P(X < 1) and P(|x| < 1)
$f(x)=\frac{x^2}{18},-3<x<3$
= 0, otherwise

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Q 16Solve the Following Question.(4 Marks)4 Marks

Find the expected value, variance and standard derivation of random variable $X$ whose probability mass function $(p.m.f.)$ is given below

$X =x$	$1$	$2$	$3$
$P ( X =x)$	$\frac{1}{5}$	$\frac{2}{5}$	$\frac{2}{5}$

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Q 17Solve the Following Question.(4 Marks)4 Marks

A bakerman sells $5$ types of cakes. Profits due to the sale of each type of cake is respectively $₹ 3,₹ 2.5,₹ 2, ₹ 1.5, ₹ 1 . $ The demands for these cakes are $10 \%, 5 \%, 25 \%, 45 \%$ and $15 \%$ respectively. What is the expected profit per cake?

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Q 18Solve the Following Question.(4 Marks)4 Marks

Find the variance and standard deviation of the random variable $X$ whose probability distribution is given below:

$x$	0	1	2	3
$P ( X =x)$	$\frac{1}{8}$	$\frac{3}{8}$	$\frac{3}{8}$	$\frac{1}{8}$

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Q 19Solve the Following Question.(4 Marks)4 Marks

Given the probability density function (p.d.f.) of a continuous random variable $x$ as:
$\begin{aligned}
f(x) & =\frac{x^2}{3}, & -1<x<2 \\
& =0, & \text { otherwise }
\end{aligned}$
Determine the cumulative distribution function (c.d.f.) of $X$ and hence find
$P ( X <1), P ( X >0), P (1< X <2) \text {. }$

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Q 20Solve the Following Question.(4 Marks)4 Marks

The following is the p.d.f. (Probability Density Function) of a continuous random variable $X$ :
$\begin{aligned}
f(x) & =\frac{x}{32}, & 0& =0, & \text { otherwise }
\end{aligned}$
(a) Find the expression for c.d.f. (Cumulative Distribution Function) of $X$.
(b) Also find its value at $x=0.5$ and 9 .

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Probability Distributions question types

Probability Distributions questions

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Probability Distributions Maths - Probability Distributions

Probability Distributions question types

Probability Distributions questions

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