Need help in (continuous probability distribution) please

Question

Please explain to me how to solve questions like that? I don't understand the answers :/

Thanks in advance <3

---

Question with answers:

Answer 1

Before getting into continuous random variables, we go back to how we deal with discrete random variables.

For a discrete random variable $X$, we have a probability mass function $f_X(x)$, which tells us the probability of a certain element $x$. $$ f_X(x) = \mathbb{P}[X=x] $$ Hence, if we have the support $S$, the set of all possible values of the random variable, we have that: $$ \sum_{x \in S} f_X(x) = 1 $$ We also have the cumulative distribution function which is defined as the probability that the random variable $X$ is at most a certain value. $$ F_X(x) = \mathbb{P}[X \leq x] $$ We can find this function using the probability mass function by simply adding up probabilities: $$ F_X(x) = \sum_{x_i \leq x} \mathbb{P}[X = x_i] = \sum_{x_i \leq x} f_X(x_i) $$

Example: If we let $X$ be the random variable denoting the value of a die, the probability mass function and cumulative distribution function are:

---

For a continuous random variable $X$, we cannot assign probability mass to individual elements, since the continuity implies an infinite quantity of elements. So we have a probability density function $f_X(x)$ which tells us the relative likelihood of a certain element $x$. Individual elements have 0 probability, but we have probabilities of intervals of elements: $$ \mathbb{P}[a \leq X \leq b] = \int_a^b f_X(x) \, dx $$ We also still have the notion of the cumulative distribution function: $$ F_X(x) = \mathbb{P}[X \leq x] = \int_{-\infty}^x f_X(t) \, dt $$

Example: The variable you have in your example is an exponential random variable with parameter $\lambda = 8$. Its probability density function and cumulative distribution function look like:

Answer 2

The probability density function $f(x)$ is defined such that the probability of $x$ being between $a$ and $b$ is given by $$P(a\leq x\leq b) = \int_a^b f(x)dx $$

What is the cumulative distribution function? $F(y)$ is the probability that the outcome $x$ will be smaller or equal to $y$, meaning $$F(y) = P(x\leq y)$$ Which if you think about it is just $$P(x\leq y) = \int_{-\infty}^y f(x)dx $$ Meaning that if you want to find the cumulative distribution function (CDF) you just integrate the probability density function (PDF).

After that you are asked about the probability that an outcome will be smaller than or equal to $0.2$ (the distribution is given in hours, 12 minutes is 0.2 of an hour), and by definition that is how the CDF is defined! (at 0.2)