The following table lists the probability distribution for...

The following table lists the probability distribution for cash prizes in a lottery conducted at Lawson's Department Store. $$\begin{array}{|rc|}\hline \text { Prize (\$) } & \text { Probability } \\\hline 0 & .45 \\10 & .30 \\100 & .20 \\500 & .05 \\\hline\end{array}$$ If you buy a single ticket, what is the probability that you win: a. Exactly $\$ 100 ?$ b. At least $\$ 10 ?$ c. No more than $\$ 100 ?$ d. Compute the mean, variance, and standard deviation of this distribution.

The following table lists the probability distribution for cash prizes in a lottery conducted at Lawson's Department Store.
$$\begin{array}{|rc|}\hline \text { Prize (\$) } & \text { Probability } \\\hline 0 & .45 \\10 & .30 \\100 & .20 \\500 & .05 \\\hline\end{array}$$
If you buy a single ticket, what is the probability that you win:
a. Exactly $\$ 100 ?$
b. At least $\$ 10 ?$
c. No more than $\$ 100 ?$
d. Compute the mean, variance, and standard deviation of this distribution.

Key Concepts

Standard Deviation

The standard deviation is the square root of the variance and serves as a measure of dispersion in the same units as the original data. It reflects the typical distance of the outcomes from the mean, offering a more interpretable measure of spread in the context of the original random variable.

Variance

The variance of a discrete random variable measures the spread of its values around the mean. It is defined as the expected value of the squared deviations from the mean. Variance provides insight into how much the values of the random variable differ from the average value.

Expected Value (Mean)

The expected value (or mean) of a discrete random variable is the weighted average of all its possible values, with the weights being their corresponding probabilities. It provides a measure of the central tendency or typical value of the distribution.

Cumulative Probability

Cumulative probability refers to the probability that a random variable will take a value less than or equal to a certain threshold. It is useful for answering questions about ranges, such as 'at least' or 'no more than' a particular value, by summing the probabilities of the relevant outcomes.

Discrete Random Variable

A discrete random variable is one that can take on a countable number of distinct values. Each of these values has an associated probability, and the sum of these probabilities is equal to 1. Discrete random variables are often represented using a probability distribution.

Probability Distribution

A probability distribution is a function or a table that lists all the possible outcomes of a random variable and assigns a probability to each outcome. It serves as a comprehensive summary of how probabilities are distributed across the different potential values of the variable.

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