In a geography assignment the grade obtained is the random...

In a geography assignment the grade obtained is the random variable $X$. It has been found that students have these probabilities of getting a specific grade: A: $0.18$ B: $0.32$ C: $0.25$ D: $0.07$ E: $0.03$ F: $0.15$ Based on this, calculate the following. a. The cumulative probability distribution of $X$. b. The probability of getting a higher grade than $B$. c. The probability of getting a lower grade than $\mathrm{C}$.

In a geography assignment the grade obtained is the random variable $X$. It has been found that students have these probabilities of getting a specific grade:
A: $0.18$
B: $0.32$
C: $0.25$
D: $0.07$
E: $0.03$
F: $0.15$
Based on this, calculate the following.
a. The cumulative probability distribution of $X$.
b. The probability of getting a higher grade than $B$.
c. The probability of getting a lower grade than $\mathrm{C}$.

Key Concepts

Discrete Random Variable

A discrete random variable is one that can take on only a countable number of distinct values. In this context, the random variable represents the grade a student receives, and each grade (A, B, C, D, E, F) is associated with a specific probability, forming a probability distribution for the outcome.

Probability Mass Function

The probability mass function (PMF) assigns probabilities to each possible outcome of a discrete random variable. Here, the PMF is given as the probability of receiving each specific grade. The PMF is essential for calculating probabilities related to individual outcomes or combinations of outcomes.

Cumulative Distribution Function

The cumulative distribution function (CDF) gives the probability that the random variable is less than or equal to a certain value. It is calculated by summing the probabilities of all outcomes up to that value. In the assignment, this involves summing the individual probabilities of grades to determine the cumulative probability up to a particular grade.

Tail Probability

Tail probability refers to the probability that a random variable takes on a value that lies above or below a certain threshold. In scenarios such as calculating the probability of getting a higher grade than B or a lower grade than C, tail probability is computed by summing the appropriate probabilities from the PMF, either in the upper or lower tail of the distribution.

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