Step 1 :First, we need to calculate the mean of the distribution. The mean is calculated as \( \mu = \sum x_i * P(x_i) \).
Step 2 :Given the values \( x_i = [0, 1, 2, 3] \) and the corresponding probabilities \( P(x_i) = [0.15, 0.1, 0.25, 0.5] \), we can calculate the mean as \( \mu = 2.1 \).
Step 3 :Next, we calculate the standard deviation using the formula \( \sigma = \sqrt{\sum (x_i - \mu)^2 * P(x_i)} \).
Step 4 :Substituting the values of \( x_i \), \( P(x_i) \), and \( \mu \) into the formula, we find that the standard deviation is \( \sigma = 1.0908712114635715 \).
Step 5 :Rounding to at least 2 decimal places, the standard deviation of this probability distribution is \(\boxed{1.09}\).