Table 4 is the probability table for a random variable X . Find E(X), Var(X), and the standard d

step 1 In this problem, we're going to be finding the expected value, the variance, and the standard de

Table 4 is the probability table for a random variable X ....

Table 4 is the probability table for a random variable $X .$ Find $\mathrm{E}(X), \operatorname{Var}(X),$ and the standard deviation of $X$. $$\begin{aligned}&\begin{array}{lll} \text {Table 4} \\ \hline \text { Outcome } & 0 & 1 \\\text { Probability } & \frac{1}{5} & \frac{4}{5} \\\hline\end{array}\end{aligned}$$

Table 4 is the probability table for a random variable $X .$ Find $\mathrm{E}(X), \operatorname{Var}(X),$ and the standard deviation of $X$.
$$\begin{aligned}&\begin{array}{lll} \text {Table 4} \\ \hline \text { Outcome } & 0 & 1 \\\text { Probability } & \frac{1}{5} & \frac{4}{5} \\\hline\end{array}\end{aligned}$$

Key Concepts

Probability Distribution

A probability distribution is a function that gives the probabilities of occurrence of different possible outcomes in an experiment. For random variables, this distribution details the probability associated with each outcome, providing a complete description of how the probability is allocated over the possible values.

Discrete Random Variable

A discrete random variable is one that can take on a countable number of distinct values. Its probability distribution assigns a probability to each individual outcome, which is particularly useful when dealing with events such as 0 or 1, where the outcomes are specific and finite.

Expected Value

The expected value, or mean, of a random variable is a measure of the center of its probability distribution. It is calculated by summing the products of each possible outcome with its corresponding probability, providing an average outcome value over many repeated trials.

Variance

Variance is a measure of the spread or dispersion of a random variable's probability distribution around its mean. It quantifies how much the outcomes vary from the expected value, calculated by taking the expected value of the squared deviations from the mean.

Standard Deviation

Standard deviation is the square root of the variance, providing a measure of dispersion that is in the same units as the original random variable. It is used to indicate how much the values of the random variable typically deviate from the mean, offering insight into the variability of the data.

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