A student needs to know details of a class assignment that...

A student needs to know details of a class assignment that is due the next day and decides to call fellow class members for this information. She believes that for any particular call, the probability of obtaining the necessary information is 0.40 . She decides to continue calling class members until the information is obtained. But her cell phone battery will not allow more than 8 calls. Let the random variable $\mathrm{X}$ denote the number of calls needed to obtain the information. a. Find the probability distribution of $X$. b. Find the cumulative probability distribution of $X$. c. Find the probability that at least three calls are required.

A student needs to know details of a class assignment that is due the next day and decides to call fellow class members for this information. She believes that for any particular call, the probability of obtaining the necessary information is 0.40 . She decides to continue calling class members until the information is obtained. But her cell phone battery will not allow more than 8 calls. Let the random variable $\mathrm{X}$ denote the number of calls needed to obtain the information.
a. Find the probability distribution of $X$.
b. Find the cumulative probability distribution of $X$.
c. Find the probability that at least three calls are required.

Key Concepts

Truncated Distribution

A truncated distribution limits the range of values that a random variable can take, often due to practical or physical constraints. In a truncated geometric distribution, the standard model is modified to account for a maximum number of trials, which means the process can be forced to stop even if it has not reached a success.

Probability Mass Function (PMF)

The probability mass function is a function that provides the probability that a discrete random variable is exactly equal to a particular value. For the geometric distribution, it is given by P(X = k) = (1 - p)^(k-1) * p, where p is the probability of success on each trial and k is the trial number.

Geometric Distribution

The geometric distribution models the number of independent Bernoulli trials required to achieve the first success, where each trial has the same fixed probability of success. This distribution is commonly used to analyze waiting times and processes that continue until a specific event occurs.

Cumulative Distribution Function (CDF)

The cumulative distribution function describes the probability that a discrete random variable is less than or equal to a particular value. In a geometric distribution context, it represents the probability that the first success occurs on or before a given trial and is calculated by summing the probabilities from the first trial up to that trial.

Transcript

00:01 Right now here on this problem, we're told a student needs no details for a class assignment.

00:05 She believes that the probability of obtaining the necessary information is 0 .40.

00:09 So p is 0 .40.

00:13 She continues calling until the information is obtained.

00:17 Now she will not be allowed more than 8 called.

00:20 So this means n is at most 8.

00:25 Now here on a we want to find the probability distribution of x.

00:29 Now she's going to stop as soon as she gets their correct answer.

00:34 What this means is that the probability she gets the correct answer on x is 0 .60 to the x minus 1.

00:44 In other words, how many times in a row does she have to get the wrong answer, or someone without any information, times 0 .40, because on that last one, she will get the right one.

00:57 And n is going to go from 1 to at most 8.

01:05 No.

01:05 B.

01:07 We want to find the cumulative probability distribution.

01:13 Now, f of x is the probability that x is less than equal to x, which means we need to sum from 0 to x of 0 .60 to x minus 1 times 0 .40.

01:34 I'm sorry, x goes from 1 because we start up 1.

01:37 And this is just a geometric series...

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