04 Photosynthesis 1

The Poisson Process Definition

• A counting process ${N(t), t \geq 0}$ is a Poisson Process with rate $\lambda > 0$ if it satisfies three conditions:
  • $N(0) = 0$
  • The process has independent increments
  • The number of events in any interval of length $t$ is Poisson distributed with mean $\lambda t$
    • $P{N(t+s) - N(s) = n} = e^{-\lambda t} \frac{(\lambda t)^n}{n!}, \quad n = 0, 1, \dots$ with $s, t \geq 0$

Interarrival Times Proposition

• Let ${T_n, n \geq 1}$ be the sequence of interarrival times, where $T_1$ is the time of the first event and $T_i$ is the time between event $i-1$ and event $i$ for $i \geq 2$
• The interarrival times $T_n, n \geq 1$ are independent and identically distributed exponential random variables with parameter $\lambda$.
• $P{T_1 > t} = P{N(t) = 0} = e^{-\lambda t}$
• $T_1 \sim Exp(\lambda)$
• $P{T_2 > t | T_1 = s} = P{0 \text{ events in } (s, s+t] | T_1 = s} = P{0 \text{ events in } (s, s+t]} = P{N(s+t) - N(s) = 0} = e^{-\lambda t}$ for $t > 0, s > 0$
• $T_2 \sim Exp(\lambda)$ where $T_1$ and $T_2$ are independent

Poisson Process (Alternative Definition)

• A Poisson process with rate $\lambda > 0$ satisfies three conditions:
  • $N(0) = 0$
  • The process has independent increments
  • The interarrival times are independent and identically distributed exponential random variables with parameter $\lambda$.

Example: Customer Arrivals at a Store

• Customers arrive at a store according to a Poisson process with rate $\lambda = 10$ customers per hour.
• Probability that no customers arrive between 11:00 and 11:30:
  • Define $N(t)$ as the number of arrivals up to time $t$, with $t=0$ corresponding to 11:00.
  • $P{N(0.5) = 0} = e^{-10 \cdot 0.5} = e^{-5} \approx 0.0067$
• Probability that at least two customers arrive between 11:00 and 11:45:
  • $P{N(0.75) \geq 2} = 1 - P{N(0.75) = 0} - P{N(0.75) = 1} = 1 - e^{-10 \cdot 0.75} - \frac{e^{-10 \cdot 0.75}(10 \cdot 0.75)^1}{1!} = 1 - e^{-7.5} - 7.5e^{-7.5} \approx 0.9981$

Thinning

• Events from a Poisson process with rate $\lambda$ can be one of two types: type I with probability $p$ and type II with probability $1-p$
• Let $N_1(t)$ and $N_2(t)$ be the number of type I and type II events in $[0, t]$, respectively
• $N_1(t)$ and $N_2(t)$ are independent Poisson processes with rates $\lambda p$ and $\lambda (1-p)$, respectively

Statistiques Inférence Notes

Estimation Ponctuelle

• $\hat{\theta}$ is an estimator of $\theta$ if it is used to estimate the value of $\theta$
• $\hat{\theta}$ is an estimate of $\theta$ if it is a particular value of $\hat{\theta}$ obtained after observing a sample

Biais (Bias)

• The bias of an estimator $\hat{\theta}$ is $Bias(\hat{\theta}) = E(\hat{\theta}) - \theta$

Erreur Quadratique Moyenne (Mean Squared Error)

• The mean squared error of an estimator $\hat{\theta}$ is $MSE(\hat{\theta}) = E[(\hat{\theta} - \theta)^2] = V(\hat{\theta}) + [Bias(\hat{\theta})]^2$

Estimateur Sans Biais (Unbiased Estimator)

• $\hat{\theta}$ is an unbiased estimator of $\theta$ if $E(\hat{\theta}) = \theta$

Estimateur Convergent (Consistent Estimator)

• $\hat{\theta}$ is a consistent estimator of $\theta$ if $\hat{\theta}$ converges in probability towards $\theta$ when $n \rightarrow \infty$

Efficacité (Efficiency)

• The most efficient estimator is the one with the lowest variance, when considering two unbiased estimators $\hat{\theta}_1$ and $\hat{\theta}_2$ of $\theta$

Information de Fisher (Fisher Information)

• Fisher Information is a measure of the amount of information that a random variable $X$ contains about the unknown parameter $\theta$.
• $I(\theta) = -E[\frac{\partial^2}{\partial \theta^2}log(L(x;\theta))]$

Borne de Cramér-Rao (Cramér-Rao Lower Bound)

• The Cramér-Rao lower bound (CRLB) is a lower bound on the variance of unbiased estimators.
• $V(\hat{\theta}) \geq \frac{1}{I(\theta)}$

Estimateur Efficace (Efficient Estimator)

• An estimator is said to be efficient if it attains the Cramér-Rao bound.

Estimation par Intervalle (Interval Estimation)

• A confidence interval (CI) is an interval in which the true value of the parameter to be estimated is expected to be found with a certain level of confidence.

Niveau de Confiance (Confidence Level)

• The confidence level is the probability that the confidence interval contains the true value of the parameter to be estimated, expressed as a percentage

Marge d'Erreur (Margin of Error)

• The margin of error is the distance between the point estimate and the bounds of the confidence interval

Interval de Confiance pour la Moyenne (Confidence Interval for the Mean)

• Variance connue (known): $\bar{X} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$, where $$\bar{X} \text{ is the sample mean}$$
  • $z_{\alpha/2} \text{ is the critical value for a } 1 - \alpha \text{ confidence level}$
  • $\sigma \text{ is the population standard deviation}$
  • $n \text{ is the sample size}$
• Variance inconnue (unknown): $\bar{X} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}}$, where
  • $\bar{X} \text{ is sample mean}$
  • $t_{\alpha/2, n-1} \text{ is the critical value of Student's t-distribution with } n-1 \text{ degrees of freedom}$
  • $s \text{ is the sample standard deviation}$
  • $n \text{ is the sample size}$

Interval de Confiance pour la Variance (Confidence Interval for the Variance)

• $\frac{(n-1)s^2}{\chi^2_{\alpha/2, n-1}} \leq \sigma^2 \leq \frac{(n-1)s^2}{\chi^2_{1-\alpha/2, n-1}}$, where
  • $s^2 \text{ is the sample variance}$
  • $\chi^2_{\alpha/2, n-1} \text{ and } \chi^2_{1-\alpha/2, n-1} \text{ are the critical values of the chi-squared distribution}$
  • $n \text{ is the sample size}$

Interval de Confiance pour une Proportion (Confidence Interval for a Proportion)

• $\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$, where
  • $\hat{p} \text{ is the sample proportion}$
  • $z_{\alpha/2} \text{ is the critical value of the standard normal distribution}$
  • $n \text{ is the sample size}$

Tests d'Hypothèses (Hypothesis Testing)

• A hypothesis test is a decision rule to choose between two statistical hypotheses (null and alternative)

Hypothèse Nulle (Null Hypothesis)

• The null hypothesis, denoted $H_0$, is the hypothesis to be refuted.

Hypothèse Alternative (Alternative Hypothesis)

• The alternative hypothesis, denoted $H_1$, is the hypothesis accepted if the null hypothesis is rejected.

Erreurs de Type I et II (Type I and Type II Errors)

• Type I Error ($\alpha$): Rejecting $H_0$ when it is true (false positive).
• Type II Error ($\beta$): Not rejecting $H_0$ when it is false (false negative).

Niveau de Signification (Significance Level)

• The significance level $\alpha$ is the maximum probability of making a Type I error.

Puissance du Test (Power of the Test)

• The power of the test is the probability of rejecting $H_0$ when $H_0$ is false. It is equal to $1 - \beta$.

p-value

• The p-value is the probability of obtaining a result at least as extreme as the one observed, assuming that the null hypothesis is true.
• If the p-value is less than $\alpha$, reject the null hypothesis.
• If the p-value is greater than $\alpha$, do not reject the null hypothesis.

Tests Paramétriques Courants (Common Parametric Tests)

• Z-test: Compares sample mean to population mean with known standard deviation.
• T-test: Compares sample mean to population mean with unknown standard deviation, or compares means of two samples
• Chi-squared test: Tests independence between categorical variables or goodness-of-fit.
• ANOVA: Compares means of multiple groups.

Tests Non Paramétriques Courants (Common Non-parametric Tests)

• Wilcoxon Test: Compares two paired samples when data does not follow a normal distribution.
• Mann-Whitney Test: Compares two independent samples when data does not follow a normal distribution
• Kruskal-Wallis Test: Compares multiple independent samples when data does not follow a normal distribution

Comparaison de nombres complexes :

Forme algébrique :

• tout nombre complexe z peut s'écrire de manière unique sous la forme z = a + ib, avec a et b des nombres réels;
• a est la partie réelle de z, notée Re(z);
• b est la partie imaginaire de z, notée Im(z);
• Egalité : deux nombres complexes sont égaux si et seulement si ils ont la même partie réelle et la même partie imaginaire
  • a + ib = a' + ib' équivaut à a = a' et b = b'

Module et argument:

• Module : le module d'un nombre complexe z = a + ib est le nombre réel positif, noté |z|, défini par : |z| = √(a^2 + b^2);
• Argument : tout nombre complexe non nul z peut s'écrire sous la forme:
  • z = |z|(cos(θ) + isin(θ));
  • θ est l'argument de z, noté arg(z); il est défini à 2π près.

Forme trigonométrique:

• Définition : la forme trigonométrique d'un nombre complexe non nul z est :
  • z = |z|(cos(θ) + isin(θ));
  • où |z| est le module de z et θ est un argument de z;
• Egalité : deux nombres complexes écrits sous forme trigonométrique sont égaux si et seulement si ils ont le même module et le même argument (à 2π près)
  • |z|(cos(θ) + isin(θ)) = |z'|(cos(θ') + isin(θ')) équivaut à |z| = |z'| et θ = θ' [2π]

Tensiones:

• la tension est une force de tracción, ejercida por una cuerda, cable, cadena o similar sobre un objeto;
• la tensión es una fuerza;
• la tensión se dirige a lo largo de la cuerda y tira igualmente en ambos extremos;
• la tensión considera que las cuerdas tienen masa despreciable y no se estiran;
• en un diagrama de cuerpo libre, la tensión se representa como un vector que apunta lejos del objeto, en la dirección de la cuerda.

Ejemplo:

• un bloque de peso w está suspendido de una cuerda;
  • Dibuja un diagrama de cuerpo libre para el bloque;
    • la tensión T tira hacia arriba;
    • el peso w tira hacia abajo;
  • aplica la segunda ley de Newton:
    • Σ Fy = may;
      • T - w = 0;
      • T = w;
        • conclusión: la tensión en la cuerda es igual al peso del bloque;

Ejemplo:

• un bloque de peso w es arrastrado a velocidad constante por una cuerda que forma un ángulo θ con la horizontal;
  • el coeficiente de fricción cinética entre el bloque y la superficie es μk;
    • Dibuja un diagrama de cuerpo libre para el bloque;
      • la tensión T tira hacia arriba y hacia la derecha;
      • el peso w tira hacia abajo;
      • la fuerza normal n empuja hacia arriba;
      • la fricción fk empuja hacia la izquierda;
    • Dibuja un diagrama de cuerpo libre para el bloque, mostrando las componentes de la tensión;
      • la tensión T se descompone en una componente horizontal Tx = T cos θ y una componente vertical Ty = T sin θ;
    • aplica la segunda ley de Newton:
      • Σ Fx = max
        • T cos θ - fk = 0;
      • Σ Fy = may
        • T sin θ + n - w = 0;
      • resuelve para la tensión
        • Como fk = μk n, tenemos:
          • T cos θ = μk n;
          • T sin θ + n = w;
            • Despejando n de la segunda ecuación y sustituyendo en la primera, obtenemos: - T cos θ = μk (w - T sin θ); - T (cos θ + μk sin θ) = μk w; - T = (μk w)/(cos θ + μk sin θ); - conclusión: la tensión en la cuerda es igual a (μk w)/(cos θ + μk sin θ);

Ejemplo:

• dos bloques están conectados por una cuerda que pasa por una polea sin fricción;
  • el bloque 1 tiene una masa m1 y el bloque 2 tiene una masa m2;
    • Dibuja un diagrama de cuerpo libre para cada bloque.
      • Bloque 1:
        • la tensión T tira hacia arriba;
        • el peso m1g tira hacia abajo;
      • Bloque 2:
        • la tensión T tira hacia arriba;
        • el peso m2g tira hacia abajo;
    • aplica la segunda ley de Newton a cada bloque:
      • Bloque 1:
        • Σ Fy = may
          • T - m1g = m1a;
      • Bloque 2:
        • Σ Fy = may
          • T - m2g = -m2a;
    • resuelve para la aceleración y la tensión
      • restando las dos ecuaciones, obtenemos:
        • m2g - m1g = m1a + m2a;
          • a = (m2 - m1)/(m1 + m2) g;
            • sustituyendo en la primera ecuación, obtenemos: - T = m1g + m1a = m1g + m1 (m2 - m1)/(m1 + m2) g; - T = (2 m1 m2)/(m1 + m2) g; - la aceleración de los bloques es (m2 - m1)/(m1 + m2) g y la tensión en la cuerda es (2 m1 m2)/(m1 + m2) g.

Fonction logarithme népérien :

Définition:

• la fonction logarithme népérien, notée ln, est définie sur ]0; +∞[ et est la primitive de la fonction x ↦ 1/x qui s'annule en 1.

Propriétés:

• Propriétés algébriques
  • pour tous réels strictement positifs a et b et pour tout entier relatif n:
    • ln(1) = 0
    • ln(e) = 1
    • ln(ab) = ln(a) + ln(b)
    • ln(a/b) = ln(a) - ln(b)
    • ln(1/b) = -ln(b)
    • ln(an) = n · ln(a)
    • ln(√a) = 1/2 ln(a)
• Étude de la fonction
  • La fonction ln est dérivable sur ]0; +∞[ et sa dérivée est la fonction x ↦ 1/x, c'est-à-dire: (ln(x))' = 1/x
    • La fonction ln est strictement croissante sur ]0; +∞[
    • lim[x→+∞] ln(x) = +∞
    • lim[x→0] ln(x) = -∞
• Courbe représentative:
  • see graph
• Dérivées:
  • Soit u une fonction dérivable et strictement positive sur un intervalle I, alors la fonction ln(u) est dérivable sur I et on a:
    • (ln(u))' = u'/u
• Limites à connaître:
  • lim[x→0] (ln(1+x))/x = 1
    • lim[x→+∞] (ln(x))/x = 0
    • lim[x→+∞] (ln(x))/x^n = 0
    • lim[x→0] x · ln(x) = 0
    • lim[x→0] x^n · ln(x) = 0

Advanced Calculus Vector Algebra

Vectors in Euclidean Space

• A vector has magnitude and direction.
  • $\mathbf{a} = (a_1, a_2,..., a_n)$ where $a_i$ are the components of the vector in $\mathbb{R}^n$
• Magnitude (length, norm) is $||\mathbf{a}|| = \sqrt{a_1^2 + a_2^2 +... + a_n^2}$
• Direction* described by the angles it makes with the coordinate axes.

Vector Operations

• Addition: $\mathbf{a} + \mathbf{b} = (a_1 + b_1, a_2 + b_2,..., a_n + b_n)$
• Scalar Multiplication: $c\mathbf{a} = (ca_1, ca_2,..., ca_n)$

Properties of Vector Operations

• Given vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ and scalars $c, d$:
• $\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}$ (Commutativity)
• $(\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c})$ (Associativity)
• $c(\mathbf{a} + \mathbf{b}) = c\mathbf{a} + c\mathbf{b}$ (Distributivity)
• $(c + d)\mathbf{a} = c\mathbf{a} + d\mathbf{a}$ (Distributivity)
• $c(d\mathbf{a}) = (cd)\mathbf{a}$ (Associativity)

Dot Product

• The dot product of two vectors $\mathbf{a}$ and $\mathbf{b}$ is:
  • $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 +... + a_nb_n$
  • or $\mathbf{a} \cdot \mathbf{b} = ||\mathbf{a}|| \cdot ||\mathbf{b}|| \cos{\theta}$, where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$

Properties of Dot Product

• $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$ (Commutativity)
• $\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}$ (Distributivity)
• $(c\mathbf{a}) \cdot \mathbf{b} = c(\mathbf{a} \cdot \mathbf{b})$ (Associativity)
• $\mathbf{a} \cdot \mathbf{a} = ||\mathbf{a}||^2 \geq 0$, and $\mathbf{a} \cdot \mathbf{a} = 0$ if and only if $\mathbf{a} = \mathbf{0}$

Orthogonality

• Two vectors $\mathbf{a}$ and $\mathbf{b}$ are orthogonal if $\mathbf{a} \cdot \mathbf{b} = 0$.

Cross Product (in $\mathbb{R}^3$)

• The cross product of $\mathbf{a} = (a_1, a_2, a_3)$ and $\mathbf{b} = (b_1, b_2, b_3)$ in $\mathbb{R}^3$ is:
• $\mathbf{a} \times \mathbf{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)$
  • or $\mathbf{a} \times \mathbf{b} = ||\mathbf{a}|| \cdot ||\mathbf{b}|| \sin{\theta} \mathbf{n}$, where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$, and $\mathbf{n}$ is a unit vector perpendicular to both

Properties of Cross Product

• $\mathbf{a} \times \mathbf{b} = -\mathbf{b} \times \mathbf{a}$ (Anti-commutativity)
• $\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}$ (Distributivity)
• $(c\mathbf{a}) \times \mathbf{b} = c(\mathbf{a} \times \mathbf{b})$ (Associativity)
• $\mathbf{a} \times \mathbf{a} = \mathbf{0}$
• $\mathbf{a} \times \mathbf{b}$ is orthogonal to both $\mathbf{a}$ and $\mathbf{b}$.

Geometric Interpretation

• The magnitude of $\mathbf{a} \times \mathbf{b}$ is equal to the area of the parallelogram formed by $\mathbf{a}$ and $\mathbf{b}$.

Scalar Triple Product

• The scalar triple product of vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ is:
  • $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$
• Represents the volume of the parallelepiped formed by the three vectors.
  • $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \end{vmatrix}$

Vector Triple Product

• The vector triple product of vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ is:
  • $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$