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

AI Thread Summary
The discussion centers on understanding the computation of the cumulative distribution function \( F_{X_1}(x) \) using the binomial distribution. The formula \( F_{X_1}(x) = \sum_{j=1}^n \binom{n}{1} F^1(x) (1-F(x))^{n-1} \) represents the probability of obtaining at least one success in \( n \) trials. It is derived from the complement rule, where \( F_{X_1}(x) \) is calculated as \( 1 - (1 - F(x))^n \). This indicates that the total probability for a specific value \( x \) sums to 1, confirming that \( F_{X_0}(x) = 1 \). The explanation emphasizes the relationship between the binomial distribution and cumulative probabilities.
WMDhamnekar
MHB
Messages
376
Reaction score
28
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.
 
Mathematics news on Phys.org
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 $F_{X_1}(x)$.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top