Show that the following function satisfies the properties of...

Show that the following function satisfies the properties of a joint probability mass function: $$ \begin{array}{llc} \hline x & y & f(x, y) \\ \hline 0 & 0 & 1 / 4 \\ 0 & 1 & 1 / 8 \\ 1 & 0 & 1 / 8 \\ 1 & 1 & 1 / 4 \\ 2 & 2 & 1 / 4 \\ \hline \end{array} $$ Determine the following: (a) $P(X < 0.5, Y < 1.5)$ (b) $P(X \leq 1)$ (c) $P(X < 1.5)$ (d) $P(X > 0.5, Y < 1.5)$ (e) Determine $E(X), E(Y), V(X),$ and $V(Y)$. (f) Marginal probability distribution of the random variable $X$ (g) Conditional probability distribution of $Y$ given that $X=1$ (h) $E(Y \mid X=1)$ (i) Are $X$ and $Y$ independent? Why or why not? (j) Calculate the correlation between $X$ and $Y$

Show that the following function satisfies the properties of a joint probability mass function:
$$
\begin{array}{llc}
\hline x & y & f(x, y) \\
\hline 0 & 0 & 1 / 4 \\
0 & 1 & 1 / 8 \\
1 & 0 & 1 / 8 \\
1 & 1 & 1 / 4 \\
2 & 2 & 1 / 4 \\
\hline
\end{array}
$$
Determine the following:
(a) $P(X < 0.5, Y < 1.5)$
(b) $P(X \leq 1)$
(c) $P(X < 1.5)$
(d) $P(X > 0.5, Y < 1.5)$
(e) Determine $E(X), E(Y), V(X),$ and $V(Y)$.
(f) Marginal probability distribution of the random variable $X$
(g) Conditional probability distribution of $Y$ given that $X=1$
(h) $E(Y \mid X=1)$
(i) Are $X$ and $Y$ independent? Why or why not?
(j) Calculate the correlation between $X$ and $Y$

Key Concepts

Probability of Events

To compute the probability of any event involving the random variables, one sums the joint probabilities over all pairs (x, y) that satisfy the event’s condition. This approach is fundamental in probability theory, where events are often defined as subsets of the sample space and one must aggregate across the applicable outcomes.

Covariance and Correlation

Covariance quantifies the degree to which two random variables change together, while correlation standardizes this measure to provide a dimensionless index between -1 and 1. A positive covariance or correlation indicates that as one variable increases, the other tends to increase; a negative value suggests that as one increases, the other tends to decrease. These concepts are essential for understanding the linear relationship between random variables.

Independence

Two random variables are said to be independent if the joint probability mass function factors into the product of the marginal probability mass functions for all outcomes. This means that knowing the value of one variable does not provide any information about the probability distribution of the other, which has significant implications in both theoretical and applied probability.

Variance

Variance is a measure of the spread or dispersion in a random variable's distribution. It quantifies how much the values of the variable deviate from the expected value on average. The variance is computed as the difference between the expected value of the square of the random variable and the square of its expected value.

Joint Probability Mass Function

A joint probability mass function (joint PMF) defines the probability distribution for two discrete random variables simultaneously. It must satisfy two key properties: the probability values must be nonnegative for all outcomes, and the sum of the probabilities over the entire support must equal one. These conditions ensure that the function is a valid probability distribution over a two-dimensional discrete space.

Conditional Probability Distribution

A conditional probability distribution describes the probability of one random variable given that another random variable takes on a specific value. It is calculated by dividing the joint probability by the marginal probability for the given conditioning event, provided that the marginal probability is not zero. This concept is central when exploring dependent behavior or relationships between random variables.

Marginal Probability Distribution

The marginal probability distribution of a random variable is obtained by summing the joint probabilities over all values of the other variable. This process ‘marginalizes’ out the unwanted variable, giving the probability distribution for the variable of interest. It is a key concept when dealing with joint distributions, particularly in problems involving multiple random variables.

Expected Value

The expected value, or mean, of a random variable is a measure of its central tendency. It is computed by summing the products of each possible value of the random variable and its corresponding probability. This measure provides an overall average outcome for the random variable and is foundational in statistical analysis.

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