How to find the mean median standard deviation of a discrete random variable

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To find the mean, median, and standard deviation of a discrete random variable, you typically need its probability distribution, which lists each possible value of the variable and its corresponding probability.

1. Mean (Expected Value)

The mean, also known as the expected value $E(X)$, is the weighted average of all possible values of the random variable, where the weights are their probabilities.

Step 1: List all possible values of the random variable $X$ (denoted as $x_i$) and their corresponding probabilities $P(X=x_i)$. Step 2: Multiply each value $x_i$ by its probability $P(X=x_i)$. Step 3: Sum these products.

The formula for the mean is: $E(X) = \mu = \sum_{i} x_i P(X=x_i)$

2. Median

The median of a discrete random variable is the smallest value $x_k$ such that the cumulative probability $P(X \le x_k)$ is greater than or equal to 0.5.

Step 1: List all possible values of $X$ in ascending order. Step 2: Calculate the cumulative probability for each value, $F(x_k) = P(X \le x_k)$. Step 3: Identify the smallest value $x_k$ for which $F(x_k) \ge 0.5$. This $x_k$ is the median.

3. Standard Deviation

The standard deviation measures the spread or dispersion of the values around the mean. It is the square root of the variance.

Step 1: Calculate the mean $E(X)$ (as described above). Step 2: Calculate the variance $Var(X)$. There are two common formulas: * Using the definition: $Var(X) = \sum_{i} (x_i - \mu)^2 P(X=x_i)$ For each value $x_i$, subtract the mean $\mu$, square the result, and multiply by its probability $P(X=x_i)$. Sum these values. * Using the computational formula (often easier): $Var(X) = E(X^2) - (E(X))^2$ First, calculate $E(X^2) = \sum_{i} x_i^2 P(X=x_i)$. This means squaring each $x_i$, multiplying by its probability, and summing. Then, subtract the square of the mean $(E(X))^2$. Step 3: Take the square root of the variance to find the standard deviation. $\sigma_X = \sqrt{Var(X)}$

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