Math Help: Understand How to Compute $F_{X_1}(x)$

In summary, the author uses the binomial distribution formula with parameters $p=F(x)$ and $n$ to calculate the probability mass function for a specific $x$. This implies that all possibilities for that $x$ add up to $1$. By using the complement rule, $F_{X_1}(x)$ can be calculated as $1-(1-F(x))^n$.
  • #1
WMDhamnekar
MHB
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1655630667801.png


Now, I don't understand how did author compute $F_{X_1}(x) = \displaystyle\sum_{j=1}^n \binom{n}{1} F^1(x) (1-F(x))^{n-1} = 1-(1-F(x))^n ?$ (I know L.H.S = R.H.S)

Would any member of Math help board explain me that? Any math help will be accepted.
 
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  • #2
The formula $\binom nj F^j(x)(1-F(x))^{n-j}$ is the probability mass function of the binomial distribution with parameters $p=F(x)$ and $n$.
Consequently we have that all possibilities for a specific $x$ sum up to $1$.
It implies that $F_{X_0}(x)=1$.
We can use the complement rule $P(A^c)=1-P(A)$ to calculate $F_{X_1}(x)$.
 

Related to Math Help: Understand How to Compute $F_{X_1}(x)$

1. How do I compute $F_{X_1}(x)$?

To compute $F_{X_1}(x)$, you will need to use the formula: $F_{X_1}(x) = P(X_1 \leq x)$, where $P$ represents the probability function and $X_1$ is the random variable. This formula will give you the cumulative probability of $X_1$ being less than or equal to a certain value, which is represented by $x$.

2. What is the difference between $F_{X_1}(x)$ and $f_{X_1}(x)$?

$F_{X_1}(x)$ represents the cumulative probability function, while $f_{X_1}(x)$ represents the probability density function. In other words, $F_{X_1}(x)$ gives the probability of a random variable being less than or equal to a certain value, while $f_{X_1}(x)$ gives the probability of a random variable taking on a specific value.

3. Can I use a calculator to compute $F_{X_1}(x)$?

Yes, you can use a calculator to compute $F_{X_1}(x)$, as long as you have the necessary data and know the formula to use. However, it is important to understand the concept behind the computation in order to accurately interpret the results.

4. How can I interpret the results of $F_{X_1}(x)$?

The results of $F_{X_1}(x)$ represent the cumulative probability of a random variable being less than or equal to a certain value. This can be interpreted as the likelihood of obtaining a value less than or equal to $x$ from a given distribution. The higher the value of $F_{X_1}(x)$, the more likely it is to obtain a value less than or equal to $x$.

5. Are there any limitations to using $F_{X_1}(x)$ in my calculations?

One limitation of using $F_{X_1}(x)$ is that it assumes the random variable follows a specific distribution, such as a normal distribution. If the data does not follow this distribution, the results may not be accurate. Additionally, $F_{X_1}(x)$ may not be useful if you need to calculate the probability of a specific value, as it only gives the cumulative probability of a range of values.

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