$F_X(x) = \begin{cases} 0 & x < t \\ \frac{x-t}{4-t} & t \le x \le 4 \\ 1 & x \ge 4 \end{cases}$
If the median of $X$ is $3$, then what is the value of $t$?
Understanding the Median Definition
The median ($m$) of a continuous random variable $X$ is the value where the cumulative distribution function (CDF) equals 0.5. Mathematically, this is expressed as $F_X(m) = 0.5$.
Applying Median to the CDF
We are given that the median of the random variable $X$ is $m = 3$. The CDF is defined as:
$F_X(x) = \frac{x-t}{4-t}$ for $t \le x \le 4$.
Since the median is 3, we know that $F_X(3) = 0.5$. We substitute $x=3$ into the CDF formula:
$F_X(3) = \frac{3-t}{4-t}$
Solving for Parameter t
Now, we set the CDF value equal to 0.5 and solve for $t$:
- Set up the equation: $\frac{3-t}{4-t} = 0.5$
- Multiply both sides by $(4-t)$ to eliminate the denominator: $3-t = 0.5(4-t)$
- Distribute the 0.5 on the right side: $3-t = 2 - 0.5t$
- Rearrange the terms to group $t$ on one side and constants on the other: $3 - 2 = t - 0.5t$
- Simplify the equation: $1 = 0.5t$
- Solve for $t$: $t = \frac{1}{0.5} = 2$
The condition for this part of the CDF is $t \le x \le 4$. Since our calculated $t=2$ and the median $x=3$, the condition $2 \le 3 \le 4$ holds true.
Final Value of t
Therefore, the value of $t$ is 2.
Important Questions from Mean Median Mode
- The median of the following set of numbers: 154, 130, 144, 137, 156, 146, 138, 149, 160, 138 is:
- The elements of the dataset $\{-5,1, a, 5, b\}$ are in ascending order. If both mean and median of the dataset are equal to 3, what is the value of $b$?
- The sample average of 50 data points is 40. The updated sample average after including a new data point taking the value of 142 is ________.
The above frequency chart shows the frequency distribution of marks obtained by a set of students in an exam. From the data presented above, which one of the following is CORRECT?
- The frequency distribution of beak sizes of a bird species is symmetric but not normally distributed. If the mean value of beak size is 6 mm, standard deviation is 25 mm and kurtosis is 10, then the median is ______ mm.
