Register

GO Classes GATE DA | Probability | Weekly Quiz 5 | Question: 2

• edited ago by
287 views
1 1 vote

Let $X$ be a continuous random variable.
$X$ has a constant PDF on $(0,1]$, and another constant PDF on $(1,2]$.
The PDF of $X$ is given by: $$ f_X(x)= \begin{cases} c, & \text{for } 0<x\leq 1,\\ d, & \text{for } 1<x\leq 2,\\ 0, & \text{otherwise}. \end{cases} $$ It is known that

$$ \mathbb{P}(1\leq X\leq 2) = 2\cdot\mathbb{P}(0\leq X<1)$$

What are the values of $c$ and $d$?

  1. $c=\frac{1}{2},\quad d=\frac{1}{2}$
     
  2. $c=\frac{1}{3},\quad d=\frac{2}{3}$
     
  3. $c=\frac{2}{3},\quad d=\frac{1}{3}$
     
  4. $c=\frac{1}{4},\quad d=\frac{3}{4}$

1 Answer

2 2 votes



Since the PDF is constant over each interval, the probability over each interval is just the height times the width:

  • $\mathbb{P}(0<X \leq 1)=c \cdot 1=c$
  • $\mathbb{P}(1<X \leq 2)=d \cdot 1=d$


From the condition, $d=2 c$.
Also, the total probability must be 1 , so:
$$
c+d=1
$$

Substituting $d=2 c$ gives:
$$
c+2 c=1 \Rightarrow 3 c=1 \Rightarrow c=\frac{1}{3}, \quad d=\frac{2}{3}
$$

• edited by
Voting & Accepting an answer helps improve the community
Answer:
← Previous · GO Classes GATE DA |...Next → · GO Classes GATE DA |...
← Previous in category · GO Classes GATE DA |...Next in category → · GO Classes GATE DA |...
Position:
Show:

Related questions

5 5 votes
1 1 answer
378
378 views
GO Classes asked Jun 27, 2025
378 views
GO Classes GATE DA | Probability | Weekly Quiz 5 | Question: 1
Let $U_{1}, U_{2}, \ldots, U_{n}$ be independent and identically distributed random variables with $U_{i} \sim \operatorname{Uniform}(0,1)$.Define the random variable $Z=...
0 0 votes
1 1 answer
361
361 views
GO Classes asked Jun 27, 2025
361 views
GO Classes GATE DA | Probability | Weekly Quiz 5 | Question: 6
If $X, Y$, and $Z$ are independent random variables taking values in $\mathbb{Z}$ (i.e., only integer values), what's the probability that $X+Y=9$ and $Y-Z=5$ ?$\sum_{k \...
2 2 votes
1 1 answer
263
263 views
GO Classes asked Jun 27, 2025
263 views
GO Classes GATE DA | Probability | Weekly Quiz 5 | Question: 7
Suppose that $X$ and $Y$ are discrete random variables taking values in $\{1,2\}$, with joint distribution given by:$$\begin{array}{ll}\operatorname{Pr}[X=1, Y=1]=\frac{1...
3 3 votes
1 1 answer
451
451 views
GO Classes asked Jun 27, 2025
451 views
GO Classes GATE DA | Probability | Weekly Quiz 5 | Question: 8
The joint probability density function of $X$ and $Y$ is given by$$f(x, y)= \begin{cases}c(x+y) & \text { if } 0 \leq x, y \leq 1 \\ 0 & \text { otherwise }\end{cases}$$f...