The random variable X has the following probability...

The random variable $X$ has the following probability distribution: $$ \begin{array}{rc} \hline x & P(X=x) \\ \hline-1 & .2 \\ 0 & .3 \\ 3 & .2 \\ 4 & .2 \\ 6 & .1 \\ \hline \end{array} $$ (a) graph the PMF and the CDF of $X$ (b) explain the relationship between the PMF and the CDF (c) find $P(X \geq 2.5)$ (d) find $P(-1<X<4)$ (e) find $P(-1 \leq X \leq 4)$ (f) find $P(X<-3)$ (g) find $P(X=1)$.

The random variable $X$ has the following probability distribution:
$$
\begin{array}{rc}
\hline x & P(X=x) \\
\hline-1 & .2 \\
0 & .3 \\
3 & .2 \\
4 & .2 \\
6 & .1 \\
\hline
\end{array}
$$
(a) graph the PMF and the CDF of $X$
(b) explain the relationship between the PMF and the CDF
(c) find $P(X \geq 2.5)$
(d) find $P(-1<X<4)$
(e) find $P(-1 \leq X \leq 4)$
(f) find $P(X<-3)$
(g) find $P(X=1)$.

Key Concepts

Discrete Random Variable

A discrete random variable is a variable that can take on a countable number of distinct outcomes, each with an associated probability. The probabilities of all possible outcomes sum up to one. Discrete random variables are used in scenarios where outcomes are distinct and separate, such as counts or specific events.

Probability Mass Function (PMF)

The PMF is a function that gives the probability that a discrete random variable is exactly equal to a specific value. It is defined for each possible outcome of the random variable and provides an intuitive way to understand the distribution of probabilities over the set of values. Graphically, the PMF is often represented by stem plots or bar charts showing the probability at each discrete value.

Cumulative Distribution Function (CDF)

The CDF of a discrete random variable is a function that calculates the cumulative probability that the variable will take a value less than or equal to a given number. It is obtained by summing the PMF values for all outcomes up to and including that number. The CDF is a non-decreasing step function that approaches 1 as the value increases over all outcomes.

Relationship Between PMF and CDF

The CDF is derived from the PMF by summing the probabilities of all outcomes that are less than or equal to a certain value. Essentially, while the PMF provides the likelihood of exact outcomes, the CDF provides cumulative probabilities, making it useful for computing probabilities over intervals or ranges of values. This relationship is central in understanding how changes or shifts in the PMF reflect in the corresponding CDF.

Probability Computation Using PMF/CDF

When calculating probabilities over intervals or for inequalities, one can either sum the individual probabilities using the PMF or use the corresponding differences in the CDF values. For instance, to find the probability that a random variable lies within a range, one sums the PMF over that range or computes the difference between the CDF values at the endpoints. This method is essential for understanding and solving probability queries involving discrete random variables.

Graphical Representation of Distributions

Graphing both the PMF and CDF provides visual insights into the behavior of a discrete random variable. The PMF is typically displayed using bar graphs or stem plots to show individual probabilities, while the CDF is depicted as a step function that jumps at the points where there is a nonzero probability. These visual tools are important for comprehending distribution characteristics, trends, and the relationship between individual outcomes and cumulative probabilities.

Transcript

00:03 In this problem, the probability distribution of the random variable x is given.

00:09 We have to determine the probabilities asked.

00:13 The first probability asked is probability of x less than or equal to 3.

00:26 In the given table, we can see that x less than or equal to 3 means either x is equal to 2 or x is equal to 3.

00:39 So, probability of x less than or equal to 3 is equal to probability of x equal to 2 plus probability of x equal to 3.

00:58 This is equal to probability of x equal to 2 is 0 .2 plus probability of x equal to 3 is 0 .4.

01:12 This is equal to 0 .6.

01:21 So probability of x less than or equal to 3 is 0 .6.

01:29 Next, it is asked to find probability of x greater than 2 .5.

01:45 In the given table we can see that x greater than 2 .5 means x is either 3, 5 or 8.

01:55 So, probability of x greater than 2 .5 is probability of x equal to 3 plus probability of x equal to 5 plus probability of x equal to 8.

02:35 This is equal to probability of x equal to 3 is 0 .4, probability of x equal to 5 is 0 .3 and probability of x equal to 8 is 0 .1.

02:52 So this is equal to 0 .4 plus 0 .3 plus 0 .1.

03:12 This is equal to 0 .8.

03:18 So probability of x greater than 2 .5 is 0 .8.

03:25 Next, it is asked probability of x lying between 2 .7 and 5 .5.

03:45 That is probability of 2 .7 less than x less than 5 .1...

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