Question

Let x denote the number of missing pieces in a randomly selected 500-piece puzzle from a local thrift shop. The variable x has the following probability distribution:

x 0 1 2 3 4
P(X) 0. 33 0. 44 0. 2 0. 02 0. 01
Calculate and interpret σx

Answer 1

The standard deviation (σx) of the random variable x is approximately 0.951.

Step-by-step explanation:

To calculate the standard deviation (σx) of the random variable x, we need to use the formula: σx = √(E[(x - μ)²]), where μ is the mean of x. In this case, the mean μ is calculated by multiplying each value of x by its corresponding probability, summing up these values, and dividing by the total number of values. So, μ = (0 * 0.33) + (1 * 0.44) + (2 * 0.2) + (3 * 0.02) + (4 * 0.01) = 0.33 + 0.44 + 0.4 + 0.06 + 0.04 = 1.27. Now, we can calculate σx by substituting the values of x, μ, and their corresponding probabilities into the formula.

σx = √((0 - 1.27)² * 0.33 + (1 - 1.27)² * 0.44 + (2 - 1.27)² * 0.2 + (3 - 1.27)² * 0.02 + (4 - 1.27)² * 0.01) = √((1.6249 * 0.33) + (0.0529 * 0.44) + (0.8425 * 0.2) + (4.6889 * 0.02) + (8.4069 * 0.01)) = √(0.5354 + 0.0232 + 0.1685 + 0.0938 + 0.0841) = √(0.905) = 0.951.

Therefore, the standard deviation (σx) of the random variable x is approximately 0.951.