Three tables listed below show "random variables" and their...

Three tables listed below show "random variables" and their "probabilities." However, only one of these is actually a probability distribution. a. Which is it? $$\begin{array}{|rc|}\hline {\boldsymbol{X}} & \boldsymbol{P}(\boldsymbol{x}) \\\hline 5 & .3 \\10 & .3 \\15 & .2 \\20 & .4 \\\hline\end{array}$$ $$\begin{array}{|rc|}\hline {x} & P(x) \\\hline 5 & .1 \\10 & .3 \\15 & .2 \\20 & .4 \\\hline\end{array}$$ $$\begin{array}{|rr|}\hline {\boldsymbol{x}} & \boldsymbol{P}(\boldsymbol{x}) \\\hline 5 & .5 \\10 & .3 \\ 15 & -.2 \\20 & .4 \\\hline\end{array}$$ b. Using the correct probability distribution, find the probability that $x$ is: (1) Exactly 15 . (2) No more than 10 . (3) More than 5 . c. Compute the mean, variance, and standard deviation of this distribution.

Three tables listed below show "random variables" and their "probabilities." However, only one of these is actually a probability distribution.
a. Which is it?
$$\begin{array}{|rc|}\hline {\boldsymbol{X}} & \boldsymbol{P}(\boldsymbol{x}) \\\hline 5 & .3 \\10 & .3 \\15 & .2 \\20 & .4 \\\hline\end{array}$$
$$\begin{array}{|rc|}\hline {x} & P(x) \\\hline 5 & .1 \\10 & .3 \\15 & .2 \\20 & .4 \\\hline\end{array}$$
$$\begin{array}{|rr|}\hline {\boldsymbol{x}} & \boldsymbol{P}(\boldsymbol{x}) \\\hline 5 & .5 \\10 & .3 \\
15 & -.2 \\20 & .4 \\\hline\end{array}$$
b. Using the correct probability distribution, find the probability that $x$ is:
(1) Exactly 15 .
(2) No more than 10 .
(3) More than 5 .
c. Compute the mean, variance, and standard deviation of this distribution.

Key Concepts

Probability Distribution

A probability distribution for a discrete random variable assigns a probability to each possible outcome such that each probability is nonnegative and the sum of all probabilities equals 1. This ensures that the distribution accurately reflects all possible outcomes in a probabilistic model, adhering to the fundamental axioms of probability.

Probability Mass Function

The probability mass function (PMF) is a function that gives the probability that a discrete random variable is exactly equal to some specific value. It encapsulates the distribution of probabilities across the outcomes and is crucial for calculating other statistical measures and probabilities for specific events.

Cumulative Probability

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a given number. It is typically calculated by summing the individual probabilities from the PMF for all outcomes up to and including the specified value. This concept is essential for solving problems that involve ranges or cumulative events.

Expected Value (Mean)

The expected value, or mean, of a discrete random variable is computed by taking the sum of the products of each outcome and its corresponding probability. It provides a measure of the central tendency or the long-run average of the random variable, serving as a key indicator in probability and statistics.

Variance

Variance measures the spread or dispersion of a probability distribution by taking the average of the squared differences between each outcome and the mean. It quantifies how much the outcomes tend to deviate from the expected value and is fundamental for assessing the degree of uncertainty in a distribution.

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 indicates how spread out the values of the random variable are around the mean, making it an intuitive measure of volatility and risk in probability distributions.

Transcript

00:01 So in this question we need to first define what is the correct probability distribution.

00:06 In this case what we need to see is if all the values that we have for p of x, if they are all greater or equal than zero, and if you sum them, like all the values that we have, if this sum is equal to 1.

00:21 So the only table that has these two information is the second table.

00:27 So the second table is the one that has an appropriate probability distribution.

00:35 Then using this table we are going to define what is the probability of finding exactly 15, which is the same as the probability of 15, and from the table this is 0 .2.

00:47 We also need to find here what is the probability that we have no more than 10, which means that x is less or equal than 10.

00:56 And in our table we only have two values that are less or equal than 10.

01:00 We have the probability of x being equal to 5 and the probability of x being equal to 10.

01:06 So we add these two together to get the answer 0 .4.

01:12 Then we need to find here the last one for item b, which is what is the probability that x is more than 5.

01:23 To say that it's more than 5 we have all these possibilities.

01:27 We have x equals to 10, x equals to 15, and x equals to 20, which can also be written as 1 minus the probability of x being less or equal than 5.

01:43 Because if you know that this sum is 0 .9, we have the 1 minus, and we only have one value that is less or equal than 5, which is being equal to 5 exactly, and the probability is 0 .1...

Please give Ace some feedback