Question

Consider a discrete random variable \( X \) that takes values from the set \( S = \{0, 1, 2, 3\} \), being the number of individuals of a species within a habitat. Consider the probability distribution of \( X \) with \( Pr(X = 0) = 0.15 \), \( Pr(X = 1) = 0.25 \), and \( Pr(X = 3) = 0.5 \), where \( Pr \) denotes probability. The value of \( Pr(X = 2) \) is \_\_\_\_\_\_\_\_. (Round off to two decimal places).

Hint:

For probability distributions:
1. Verify that the sum of all probabilities equals 1.
2. Use this property to find missing probabilities by subtracting the sum of known values from 1.

Answer 1

Step 1: Sum of probabilities in a probability distribution.
For any discrete random variable, the sum of probabilities of all possible outcomes must equal 1: \[ Pr(X = 0) + Pr(X = 1) + Pr(X = 2) + Pr(X = 3) = 1. \] Step 2: Substitute the known probabilities. Given: \[ Pr(X = 0) = 0.15, \quad Pr(X = 1) = 0.25, \quad Pr(X = 3) = 0.5. \] Substitute these values: \[ 0.15 + 0.25 + Pr(X = 2) + 0.5 = 1. \] Step 3: Solve for \( Pr(X = 2) \). Simplify the equation: \[ Pr(X = 2) = 1 - (0.15 + 0.25 + 0.5). \] \[ Pr(X = 2) = 1 - 0.9 = 0.10. \]