Question

If, \(X \sim \text{Bin}(8, 1/2)\) and \(Y = X^2+2\), then \(P(Y \le 6)\) is:

• A. 0.036
• B. 0.185
• C. 0.08
• D. 0.165

Hint:

When dealing with transformations of discrete random variables (\(Y = g(X)\)), always start by determining the set of X-values that correspond to the event for Y. Then, sum the probabilities of those X-values. Be careful with inequalities and remember the domain of the original variable (e.g., non-negative integers for Binomial).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem involves a transformation of a discrete random variable. We need to find the probability of an event defined for the transformed variable Y. The first step is to translate the event for Y into an equivalent event for the original variable X. Then, we use the probability mass function (PMF) of X to calculate the required probability.

Step 2: Key Formula or Approach:
1. Convert the inequality for Y into an inequality for X: \( Y \le 6 \implies X^2+2 \le 6 \). 2. Solve for the possible integer values of X. 3. Calculate the probabilities for these values of X using the Binomial PMF: \( P(X=k) = \binom{n}{k}p^k(1-p)^{n-k} \). 4. Sum the probabilities.

Step 3: Detailed Explanation:
1. Translate the event: \[ Y \le 6 \] \[ X^2 + 2 \le 6 \] \[ X^2 \le 4 \] Since X represents the number of successes, it must be a non-negative integer. The integer values of X that satisfy \(X^2 \le 4\) are \(X = 0, 1, 2\). 2. The problem now is to calculate \( P(X \le 2) = P(X=0) + P(X=1) + P(X=2) \). 3. We use the Binomial PMF with \(n=8\) and \(p=1/2\). The term \(p^k(1-p)^{n-k}\) becomes \( (1/2)^k (1/2)^{8-k} = (1/2)^8 = 1/256 \). - \( P(X=0) = \binom{8}{0} \left(\frac{1}{2}\right)^8 = 1 \times \frac{1}{256} = \frac{1}{256} \) - \( P(X=1) = \binom{8}{1} \left(\frac{1}{2}\right)^8 = 8 \times \frac{1}{256} = \frac{8}{256} \) - \( P(X=2) = \binom{8}{2} \left(\frac{1}{2}\right)^8 = \frac{8 \times 7}{2 \times 1} \times \frac{1}{256} = 28 \times \frac{1}{256} = \frac{28}{256} \) 4. Sum the probabilities: \[ P(Y \le 6) = P(X \le 2) = \frac{1}{256} + \frac{8}{256} + \frac{28}{256} = \frac{37}{256} \] 5. Convert to a decimal: \[ \frac{37}{256} \approx 0.1445 \] The calculated probability is 0.1445. This value is not exactly among the options. However, 0.165 is the closest option. It is likely there is a typo in the question or the options, but in an exam context, the closest answer would be chosen.
Step 4: Final Answer:
The calculated probability is \(37/256 \approx 0.1445\). The closest option provided is 0.165.