Vidaara.orgClass 12 · Mathematics
CodeVID-M12-13-RVD-01
Random Variables & Distributions — Assignment
Name: ____________________
Roll No.: __________
Date: ____________
General Instructions
- All questions are compulsory.
- Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
- Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions
5 × 1 = 5 marks
1.
For a valid distribution, $\sum p_i=$
- A.$0$
- B.$1$
- C.$n$
- D.$\mu$
2.
If $P(X=x)=kx$ for $x=1,2,3$, then $k=$
- A.$\tfrac16$
- B.$\tfrac13$
- C.$\tfrac12$
- D.$1$
3.
$E(X)$ is defined as:
- A.$\sum x_i$
- B.$\sum x_i p_i$
- C.$\sum p_i$
- D.$\sum x_i^2$
4.
Two coins tossed, $X=$ heads; $E(X)=$
- A.$0.5$
- B.$1$
- C.$1.5$
- D.$2$
5.
The variance is:
- A.$\sum x_i p_i$
- B.$\sum x_i^2 p_i-\mu^2$
- C.$\mu^2$
- D.$\sum p_i$
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
For a valid probability distribution, what is $\sum p_i$?
7.
If $P(X=x)=kx$ for $x=1,2,3$, find $k$.
8.
Two coins are tossed and $X$ is the number of heads. Find $P(X=2)$.
9.
Find $E(X)$ for $X:1,2$ with $p:\tfrac12,\tfrac12$.
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
Find $E(X)$ for $X:1,2,3$ with $p:\tfrac16,\tfrac26,\tfrac36$.
11.
Two coins are tossed; $X=$ number of heads. Find $E(X)$.
12.
Find $k$ if $P(X=x)=kx^2$ for $x=1,2,3$.
13.
A fair die is rolled and $X$ is the number shown. Find $E(X)$.
Section D — Long Answer (5 marks)
2 × 5 = 10 marks
14.
A random variable $X$ has distribution $X:0,1,2$ with $P:\tfrac14,\tfrac12,\tfrac14$. Find $E(X)$ and $\operatorname{Var}(X)$.
15.
Two cards are drawn from a deck. Let $X$ be the number of aces. Find the probability distribution of $X$.
Answer Key
Section A — Multiple Choice Questions
- (B) $1$
- (A) $\tfrac16$
- (B) $\sum x_i p_i$
- (B) $1$
- (B) $\sum x_i^2 p_i-\mu^2$
Section B — Short Answer (2 marks)
- $1$.
- $\tfrac16$.
- $\tfrac14$.
- $1.5$.
Section C — Short Answer (3 marks)
- $\tfrac73$.
- $1$.
- $\tfrac{1}{14}$.
- $3.5$.
Section D — Long Answer (5 marks)
- $E(X)=1,\ \operatorname{Var}(X)=0.5$.
- $P(0)=\tfrac{188}{221},\ P(1)=\tfrac{32}{221},\ P(2)=\tfrac{1}{221}$.
Generated by Vidaara.org · Assignment VID-M12-13-RVD-01 · vidaara.org