Question

(a) 10 identical blocks are marked with '0' on two of them, '1' on three of them, and '2' on four of them. If $X$ denotes the number written on the block, then write the probability distribution of $X$ and calculate its mean.

Hint:

To find the mean of a discrete random variable, multiply each possible value by its probability and sum the results.

Answer 1

The total number of blocks is 10, and the probabilities of selecting each type of block are: - Probability of selecting a block marked with '0' (2 blocks): \[ P(X = 0) = \frac{2}{10} = 0.2 \] - Probability of selecting a block marked with '1' (3 blocks): \[ P(X = 1) = \frac{3}{10} = 0.3 \] - Probability of selecting a block marked with '2' (4 blocks): \[ P(X = 2) = \frac{4}{10} = 0.4 \] Thus, the probability distribution of $X$ is: \[ P(X = 0) = 0.2, \quad P(X = 1) = 0.3, \quad P(X = 2) = 0.4 \] The mean of $X$ (expected value) is calculated as: \[ E(X) = 0 \times 0.2 + 1 \times 0.3 + 2 \times 0.4 = 0 + 0.3 + 0.8 = 1.1 \] Thus, the mean is 1.1.