Let n be a positive integer and F(x)=(⌊x⌋)/(n) for 0 ≤x ≤n, F(x)=0 for x n,

step 1 Hi everyone, so here for our answer we have f of n is equals to a minus x raised to the power of

Let n be a positive integer and F(x)=(⌊x⌋)/(n) for 0 ≤x ≤n,...

Let $n$ be a positive integer and
$$
F(x)=\frac{\lfloor x\rfloor}{n}
$$
for $0 \leq x \leq n, F(x)=0$ for $x<0$, and $F(x)=1$ for $x>n$, where $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$. Show that $F$ is a CDF, and find the PMF that it corresponds to.

Key Concepts

Cumulative Distribution Function (CDF)

A CDF is a function that, for each value x, represents the probability that a random variable takes on a value less than or equal to x. It has to satisfy certain properties such as being non-decreasing, right-continuous, approaching 0 as x goes to negative infinity and approaching 1 as x goes to positive infinity. These properties ensure that the function accurately represents a probability distribution in a cumulative manner.

Discrete Random Variables

Discrete random variables are those that have a countable number of possible outcomes. Their probability distributions are typically defined by assigning specific probabilities to each outcome. Because they take on isolated values, their cumulative distribution functions appear as step functions, with jumps at the points where the random variable has non-zero probability.

Probability Mass Function (PMF)

The PMF of a discrete random variable assigns a probability to each possible outcome. It can be derived from the CDF by computing the differences (or jumps) at the points of discontinuity. This relationship is fundamental in understanding how the total probability distributed across individual outcomes is reflected in the overall cumulative probability distribution.

Step Functions

Step functions are functions that remain constant over intervals and change value only at specific points. In the context of probability distributions for discrete random variables, the CDF is a step function because the probability mass is concentrated at distinct points. Each 'step' corresponds to an outcome with its assigned probability, making it easier to visualize and compute the probabilities associated with discrete events.

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