Question

The probability distribution of a random variable X is given by
\begin{tabular}{|c|c|c|c|} \hline X & 0 & 1 & 2 \\ \hline P(X) & $1 - 7a^2$ & $\frac{1}{2}a + \frac{1}{4}$ & $a^2$ \\ \hline \end{tabular}
If a > 0, then P(0 $<$ x $\le$ 2) is equal to

• A. $\frac{1}{16}$
• B. $\frac{3}{18}$
• C. $\frac{7}{16}$
• D. $\frac{9}{16}$

Hint:

The two fundamental properties of a discrete probability distribution are: 1) \(0 \le P(X=x_i) \le 1\) for all \(x_i\), and 2) \(\sum P(X=x_i) = 1\). The second property is almost always the starting point for finding unknown parameters.

Answer 1

The Correct Option is C

Step 1: Core Principle:
For any probability distribution, the sum of probabilities for all possible outcomes must equal 1. Mathematically, \(\sum P(X=x_i) = 1\).
Step 2: Methodology:
1. Formulate an equation by summing probabilities and setting them equal to 1.
2. Solve for the unknown parameter 'a'.
3. Utilize the determined value of 'a' to compute the desired probability.
Step 3: Calculation Breakdown:
The sum of probabilities is:
\[ P(X=0) + P(X=1) + P(X=2) = 1 \] \[ (1 - 7a^2) + \left(\frac{1}{2}a + \frac{1}{4}\right) + (a^2) = 1 \] Simplify the equation:
\[ 1 - 6a^2 + \frac{1}{2}a + \frac{1}{4} = 1 \] Subtract 1 from both sides:
\[ -6a^2 + \frac{1}{2}a + \frac{1}{4} = 0 \] Multiply by 4 to clear fractions:
\[ -24a^2 + 2a + 1 = 0 \] Multiply by -1 for a positive leading coefficient:
\[ 24a^2 - 2a - 1 = 0 \] Factor the quadratic equation:
\[ 24a^2 - 6a + 4a - 1 = 0 \] \[ 6a(4a - 1) + 1(4a - 1) = 0 \] \[ (6a + 1)(4a - 1) = 0 \] Possible values for 'a' are \(a = -\frac{1}{6}\) or \(a = \frac{1}{4}\).
Given \(a>0\), we select \(a = \frac{1}{4}\).
We need to calculate \(P(0<X \le 2)\), which is \(P(X=1) + P(X=2)\).
\[ P(0<X \le 2) = P(X=1) + P(X=2) \] \[ P(0<X \le 2) = \left(\frac{1}{2}a + \frac{1}{4}\right) + (a^2) \] Substitute \(a = \frac{1}{4}\):
\[ P(0<X \le 2) = \left(\frac{1}{2}\left(\frac{1}{4}\right) + \frac{1}{4}\right) + \left(\frac{1}{4}\right)^2 \] \[ = \left(\frac{1}{8} + \frac{1}{4}\right) + \frac{1}{16} \] \[ = \left(\frac{1}{8} + \frac{2}{8}\right) + \frac{1}{16} = \frac{3}{8} + \frac{1}{16} \] \[ = \frac{6}{16} + \frac{1}{16} = \frac{7}{16} \] Step 4: Conclusion:
The calculated probability \(P(0<x \le 2)\) is $\frac{7{16}$}.