Understanding the T-Test

The t-test

• Comparing two samples is a common task to see if they differ

• Examples include gene expression, blood pressure with drug vs. placebo, and student heights

• The null hypothesis assumes both samples come from the same distribution

• The t-test compares the means of two normally distributed samples

• Versions exist for paired/unpaired samples and equal/unequal variances

The t-statistic

• Measures the difference between sample means relative to variability within samples

• Formula: $t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$

  • $\bar{x}_1$, $\bar{x}_2$ are sample means
  • $n_1$, $n_2$ are sample sizes
  • $s_p$ is pooled standard deviation, estimating population standard deviation assuming equal variance

• Follows a t-distribution with $n_1 + n_2 - 2$ degrees of freedom

The p-value

• The probability of observing a t-statistic as extreme or more extreme than calculated
• Assumes the null hypothesis is true, calculated using the t-distribution
• Reject the null hypothesis if the p-value is less than significance level (usually 0.05)

T-test Assumptions

• Samples are normally distributed and independent
• Student's t-test assumes equal variances
• If these assumptions are not met, the test is not valid

Student's t-test

• Also known as the independent samples or unpaired t-test
• Assumes equal variance between the two samples
• Pooled standard deviation formula: $s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}$
  • $s_1^2$, $s_2^2$ are sample variances

Welch's t-test

• Also known as the unequal variance t-test
• Does not assume equal variance, and uses this t-statistic: $t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$
• Degrees of freedom are calculated with this formula: $v = \frac{(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2})^2}{\frac{(\frac{s_1^2}{n_1})^2}{n_1 - 1} + \frac{(\frac{s_2^2}{n_2})^2}{n_2 - 1}}$

Paired t-test

• Used for paired samples, such as before/after treatment measurements on the same individual
• Calculates the difference between each pair
• Performs a one-sample t-test on the differences
• t-statistic calculation: $t = \frac{\bar{d}}{s_d / \sqrt{n}}$
  • $\bar{d}$ is mean of the differences
  • $s_d$ is standard deviation of the differences
  • $n$ is the number of pairs

Alternative Non-Parametric Tests

• Mann-Whitney U test compares two samples without assuming normal distribution, based on data point ranks
• Kolmogorov-Smirnov test compares two samples without assuming normal distribution to test if they come from the same distribution, and is sensitive to location/shape differences

Key points for T-tests

• The t-test is a standard way to compare two sample means assuming normal distribution
• There are several versions of the t-test
• Other tests may be more appropriate if t-test assumptions aren't met

9.1 Linear Momentum

• Linear momentum ($\overrightarrow{\mathbf{p}}$) equals mass ($m$) times velocity ($\overrightarrow{\mathbf{v}}$): $\overrightarrow{\mathbf{p}} \equiv m \overrightarrow{\mathbf{v}}$
• Momentum is a vector, pointing in the same direction as velocity
• It has dimensions of ML/T and SI units of kg m/s

Newton's Second Law and Momentum

• The resultant external force on a system equals the time rate of change of the system's total momentum: $\sum \overrightarrow{\mathbf{F}}=\frac{d \overrightarrow{\mathbf{p}}}{d t}$

9.2 Isolated System

• The total momentum of an isolated system remains constant when two or more particles interact:
  • $\overrightarrow{\mathbf{p}}{1 i}+\overrightarrow{\mathbf{p}}{2 i}=\overrightarrow{\mathbf{p}}{1 f}+\overrightarrow{\mathbf{p}}{2 f}$
  • $m_{1} \overrightarrow{\mathbf{v}}{1 i}+m{2} \overrightarrow{\mathbf{v}}{2 i}=m{1} \overrightarrow{\mathbf{v}}{1 f}+m{2} \overrightarrow{\mathbf{v}}_{2 f}$

9.3 Impulse and Momentum

• From Newton's Second Law, $d \overrightarrow{\mathbf{p}}=\overrightarrow{\mathbf{F}} d t$
• Integrating to find momentum change over time interval $\Delta t=t_{f}-t_{i}$ gives:
  • $\Delta \overrightarrow{\mathbf{p}}=\overrightarrow{\mathbf{p}}{f}-\overrightarrow{\mathbf{p}}{i}=\int_{t_{i}}^{t_{f}} \overrightarrow{\mathbf{F}} d t$

Impulse Defined

• Impulse ($\overrightarrow{\mathbf{I}}$) is the integral of force ($\overrightarrow{\mathbf{F}}$) acting on a particle over time: $\overrightarrow{\mathbf{I}} \equiv \int_{t_{i}}^{t_{f}} \overrightarrow{\mathbf{F}} d t$
• Impulse is a vector, with its magnitude equal to the area under the Force-Time curve
• Impulse has the same units as momentum

Impulse-Momentum Theorem

• The change in a particle's momentum equals the impulse of the net force acting on it: $\Delta \overrightarrow{\mathbf{p}}=\overrightarrow{\mathbf{I}}$

Vectors in Physics

• Vectors have magnitude and direction and examples of these are: displacement, velocity, acceleration, force
• Vector notations include $\overrightarrow{A}$, $\mathbf{A}$, and $\underline{A}$

Vector Operations

Equality

• Vector $\overrightarrow{A}$ equals vector $\overrightarrow{B}$ if and only if $A_x = B_x$ and $A_y = B_y$

Addition

• $\overrightarrow{C} = \overrightarrow{A} + \overrightarrow{B}$ implies $C_x = A_x + B_x$ and $C_y = A_y + B_y$
• Vector addition is commutative: $\overrightarrow{A} + \overrightarrow{B} = \overrightarrow{B} + \overrightarrow{A}$
• Vector addition is associative: $(\overrightarrow{A} + \overrightarrow{B}) + \overrightarrow{C} = \overrightarrow{A} + (\overrightarrow{B} + \overrightarrow{C})$

Subtraction

• $\overrightarrow{C} = \overrightarrow{A} - \overrightarrow{B}$ implies $C_x = A_x - B_x$ and $C_y = A_y - B_y$

Multiplication by a Scalar

• $\overrightarrow{B} = c\overrightarrow{A}$ implies $B_x = cA_x$ and $B_y = cA_y$

Vector Components

• Vector $\overrightarrow{A}$ can be expressed as $\overrightarrow{A} = A_x\hat{i} + A_y\hat{j}$
• $A_x = A\cos\theta$
• $A_y = A\sin\theta$
• $A = \sqrt{A_x^2 + A_y^2}$
• $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$

Dot Product

• $\overrightarrow{A} \cdot \overrightarrow{B} = A B \cos\theta = A_x B_x + A_y B_y + A_z B_z$
• Dot product is commutative: $\overrightarrow{A} \cdot \overrightarrow{B} = \overrightarrow{B} \cdot \overrightarrow{A}$
• Dot product is distributive: $\overrightarrow{A} \cdot (\overrightarrow{B} + \overrightarrow{C}) = \overrightarrow{A} \cdot \overrightarrow{B} + \overrightarrow{A} \cdot \overrightarrow{C}$
• $\overrightarrow{A} \cdot \overrightarrow{A} = |\overrightarrow{A}|^2 = A^2$

Cross Product

• $\overrightarrow{A} \times \overrightarrow{B} = AB\sin\theta \hat{n}$
• $\hat{n}$ is a unit vector perpendicular to the plane formed by $\overrightarrow{A}$ and $\overrightarrow{B}$, as per right-hand rule
• $\overrightarrow{A} \times \overrightarrow{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \end{vmatrix} = (A_y B_z - A_z B_y)\hat{i} - (A_x B_z - A_z B_x)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$
• Cross product is anti-commutative: $\overrightarrow{A} \times \overrightarrow{B} = -\overrightarrow{B} \times \overrightarrow{A}$
• Cross product is distributive: $\overrightarrow{A} \times (\overrightarrow{B} + \overrightarrow{C}) = \overrightarrow{A} \times \overrightarrow{B} + \overrightarrow{A} \times \overrightarrow{C}$
• $\overrightarrow{A} \times \overrightarrow{A} = 0$

Algorithmes gloutons (Greedy Algorithms)

• Greedy algorithms offer a simple approach to optimization problems, making the best local choice at each step
• They aim to find the optimal global solution but may not always succeed

Rendre la monnaie (Making Change)

• A greedy algorithm can be used to dispense the least number of coins for a given sum:
  1. Choose the largest denomination coin less than or equal to the remaining sum.
  2. Subtract the coin's value from the remaining sum.
  3. Repeat steps 1 and 2 until the sum is zero.
• Greedy algorithms have limitations and do not guarantee optimal solutions due to their potential to get stuck in local optima.

Problème du sac à dos (Knapsack Problem)

• In the knapsack problem, a thief must choose items to maximize total value without exceeding a limited sack capacity

Greedy Approaches

1. Highest value first may not be optimal
2. Lowest weight first may not be optimal
3. Best value/weight ratio first is often best

Exemples

• The knapsack problem and its greedy approaches

Arbres couvrants minimaux (Minimum Spanning Trees)

• A spanning tree is a subgraph of a connected graph, is also a tree, and includes all original graph nodes
• A minimum spanning tree (MST/ACM) is the spanning tree with the lowest edge weight sum

Algorithme de Prim (Prim's Algorithm)

1. Pick any node as your starting point
2. Append the minimum weight edge that is connected to a node that is in a tree, and to a node that isn't
3. Repeat until all nodes have been added

Algorithme de Kruskal (Kruskal's Algorithme)

1. Order all edges from least to greatest
2. Append the smallest edged link, as long as the new edge doesn't create a cyclic tree
3. Repeat until all nodes are linked

Conclusion

• Greedy algorithms are simple and effective, but they might not get the best results in all scenarios. Understanding their constraints is key before using them

Channel Capacity

• Channel Capacity is the tightest upper bound on the rate of information that can be reliably transmitted over a channel
• $C = \max_{p(x)} I(X; Y)$, where C is channel capacity in bits per channel use, p(x) is input signal distribution, and I(X; Y) is mutual information between input (X) and output signals (Y)

Additive White Gaussian Noise (AWGN) Channel

• AWGN channels are a common model where Gaussian distributed noise is added to the signal: $Y = X + Z$
  • Y is the received signal
  • X is the transmitted signal
  • Z is the noise, following a Gaussian distribution with mean zero and variance N

Capacity of AWGN Channel

• Capacity C of an AWGN channel with bandwidth B (Hz) and Signal-to-Noise Ratio SNR (Shannon-Hartley theorem): $C = B \log_2(1 + SNR)$ where:
  • C is channel capacity (bits per second)
  • B is bandwidth of the channel (Hz)
  • $SNR = \frac{P}{N}$, is the signal-to-noise ratio, where P is average received signal power and N is average noise power

Key Points

• Bandwidth (B): Higher bandwidth enables transmission of more data per unit of time
• Signal-to-Noise Ratio (SNR): Stronger signal relative to noise enhances the reliability of communication
• The formula defines the maximum reliable transmission rate, given bandwidth and SNR
• This capacity is a fundamental limit challenging to achieve due to coding, modulation, and conditions

Information Transmission

• Shannon’s Channel Coding Theorem says reliable communication is possible if rate R is less than capacity C
  • No codes
  • Capacity is a fundamental limit

Channel Capacity

• Channel capacity of a discrete memoryless channel (DMC) is defined as $C = \max_{p(x)} I(X;Y)$
  • $I(X;Y)$ is the mutual information between the input and output of the channel.
  • The maximum is taken over all possible input distributions $p(x)$.

Examples of Channel Capacity

Noiseless Binary Channel

• Noiseless binary channel capacity is $C = 1$ bit

Noisy Channel with Nonoverlapping Outputs

• $C = H(X)$

Noisy Typewriter

• $I(X;Y) = H(Y) - H(Y|X) = H(Y) - H(Z)$
• $C = \max H(Y) - 1$
• $\max H(Y) = log(13)$
• $C = log(13) - 1$ bits

Binary Symmetric Channel (BSC)

• Probability of correct reception is 1-p, with an error probability of p
• $C = 1 - H(p)$ bits, where $H(p) = -p\log(p) - (1-p)\log(1-p)$

Binary Erasure Channel (BEC)

• $C = 1 - α$, with an erasure probability of α

Channel Capacity - Discrete Input, Continuous Output

• $C = \max_{p(x)} I(X;Y)$
• $I(X;Y) = h(Y) - h(Y|X)$
• $h(Y|X) = \sum_x p(x) h(Y|X=x)$

Binary Input, Gaussian Output

• $Y = X + Z$, where $X \in {+a, -a}$ and $Z \sim N(0, N)$
• $I(X;Y) = h(Y) - h(Y|X)$
• $h(Y|X) = h(X + Z | X) = h(Z|X) = h(Z)$
• $h(Z) = \frac{1}{2} \log(2\pi eN)$
• $C = \max_{p(x)} h(Y) - \frac{1}{2} \log(2\pi eN)$

5. 1 Preliminary Theory

Linear differential equations: initial value problems

• Definition 5.1.1: A linear differential equation of order $n$ is an equation of the form: $$ a_{n}(x) \frac{d^{n} y}{d x^{n}}+a_{n-1}(x) \frac{d^{n-1} y}{d x^{n-1}}+\cdots+a_{1}(x) \frac{d y}{d x}+a_{0}(x) y=g(x) $$

• Standard Form: Divide the equation (1) by the leading coefficient $a_{n}(x)$ to get the standard form: $$ \frac{d^{n} y}{d x^{n}}+P_{n-1}(x) \frac{d^{n-1} y}{d x^{n-1}}+\cdots+P_{1}(x) \frac{d y}{d x}+P_{0}(x) y=f(x) $$ where $P_{i}(x)=a_{i}(x) / a_{n}(x), i=0,1, \ldots, n-1$ and $f(x)=g(x) / a_{n}(x)$.

• Initial Value Problem: Find a solution to equation (1) on an interval $I$ containing $x_{0}$ that satisfies the $n$ inital conditions: $$ y\left(x_{0}\right)=y_{0}, \quad y^{\prime}\left(x_{0}\right)=y_{1}, \ldots, \quad y^{(n-1)}\left(x_{0}\right)=y_{n-1}, $$ where $y_{0}, y_{1}, \ldots, y_{n-1}$ are arbitrary real constants. Solve: $$ \begin{aligned} &a_{n}(x) \frac{d^{n} y}{d x^{n}}+a_{n-1}(x) \frac{d^{n-1} y}{d x^{n-1}}+\cdots+a_{1}(x) \frac{d y}{d x}+a_{0}(x) y=g(x) \ &y\left(x_{0}\right)=y_{0}, \quad y^{\prime}\left(x_{0}\right)=y_{1}, \ldots, \quad y^{(n-1)}\left(x_{0}\right)=y_{n-1} \end{aligned} $$

• Theorem 5.1.1 Existence of a Unique Solution: Let $P_{i}(x)$ and $f(x)$ be continuous on an interval $I$ containing $x_{0}$. Then there exists a unique solution of (2) satisfying (3) on the interval $I$.

Linear differential equations: superposition principle

• General Solution: On an interval $I$ where the $P_{i}(x)$ are continous, the general solution of the diffential equation is: $$ \frac{d^{n} y}{d x^{n}}+P_{n-1}(x) \frac{d^{n-1} y}{d x^{n-1}}+\cdots+P_{1}(x) \frac{d y}{d x}+P_{0}(x) y=f(x) $$ is given by $$ y=c_{1} y_{1}(x)+c_{2} y_{2}(x)+\cdots+c_{n} y_{n}(x)+y_{p}(x) $$ where $c_{i}, i=1, \ldots, n$ are arbitrary constants, $y_{i}(x), i=1, \ldots, n$ are linearly independent solutons of the homogeneous equation $$ \frac{d^{n} y}{d x^{n}}+P_{n-1}(x) \frac{d^{n-1} y}{d x^{n-1}}+\cdots+P_{1}(x) \frac{d y}{d x}+P_{0}(x) y=0 $$ and $y_{p}(x)$ is a particaulr soluton of the nonhomogeneous equation (2).

• *Homogeneous Equation*: If the $g(x)=0$, the equation (1) becomes
  

  $$ a_{n}(x) \frac{d^{n} y}{d x^{n}}+a_{n-1}(x) \frac{d^{n-1} y}{d x^{n-1}}+\cdots+a_{1}(x) \frac{d y}{d x}+a_{0}(x) y=0 . $$

• Trivial Solutions: The soluton $y=0$ is a soluton of the homogeneous equation.

• Definition 5.1.2: A set of functions $f_{1}(x), f_{2}(x), \ldots, f_{n}(x)$ is said to be linearly dependent on an interval $I$ if there exist constants $c_{1}, c_{2}, \ldots, c_{n}$, not all zero, such that $$ c_{1} f_{1}(x)+c_{2} f_{2}(x)+\cdots+c_{n} f_{n}(x)=0 $$ for all $x$ in the interval $I$. If the set of functions is not linearly dependent, it is said to be linearly independent.

Regulation of Gene Expression - The Lac Operon

• What is it? It's a functional genetic unit, made of structural genes, control elements (promoter & operator), and regulatory genes
• Function: The Lac operon allows the E. coli bacterium to use lactose as an energy source, which is useful if glucose is not available
• How It Works: Expression is controlled by presence/absence of lactose or glucose

Components of the Lac Operon

• Structural Genes:
• lacZ: Encodes $\beta$-galactosidase, which breaks down lactose into glucose and galatose
• lacY: Encodes lactose permease, which allows to cell to bring lactose in
• lacA: Transacetylase, takes acetyl groups and moves them
• Control Elements:
• Promotor (P): RNA polymerase binds there
• Operator (O): Region where the protein repressor bonds.
• Regulatory Gene (lacI):
• Codes for a repressor, preventing transcription

How the Lac Operon is Regulated

1. No Lactose

• Repressor is active and bonded to the operator, which will not allow transcription to occur

2. Lactose is Present

• Lactose's isomer, allolactose, deactivates the protein repressor. The RNA can now bind, and transcription can occur

3. Glucose Effect (Catabolic Repression)

• Low Glucouse
• High cAMP levels, favors CAP
• Transcription rises

Summary of Lac Operon

PhysicsKinematics

MRU

$x_f = x_i + v(t_f - t_i)$

MRUV

• $v_f = v_i + a(t_f - t_i)$
• $x_f = x_i + v_i(t_f - t_i) + \frac{1}{2}a(t_f - t_i)^2$
• $v_f^2 = v_i^2 + 2a(x_f - x_i)$

Lliure Fall

The equations for the free fall are:

• $v_f = v_i - g(t_f - t_i)$
• $y_f = y_i + v_i(t_f - t_i) - \frac{1}{2}g(t_f - t_i)^2$
• $v_f^2 = v_i^2 - 2g(y_f - y_i)$

Parabolic Shot

$v_{0x} = v_0cos(\theta)$ $v_{0y} = v_0sen(\theta)$ $v_{fx} = v_{0x}$ $x_f = x_i + v_{0x}(t_f - t_i)$ $v_{fy} = v_{0y} - g(t_f - t_i)$ $y_f = y_i + v_{0y}(t_f - t_i) - \frac{1}{2}g(t_f - t_i)^2$ $t_{total} = \frac{2v_{0y}}{g}$ $x_{max} = \frac{v_0^2sen(2\theta)}{g}$ $y_{max} = \frac{v_{0y}^2}{2g}$

MCU

$\omega = \frac{\Delta \theta}{\Delta t}$ $v = \omega r$ $a_c = \frac{v^2}{r} = \omega^2r$ $T = \frac{2\pi r}{v} = \frac{2\pi}{\omega}$ $f = \frac{1}{T}$

Dynamics

• Newton's Laws* $\sum \overrightarrow{F} = m\overrightarrow{a}$ $F_{AB} = -F_{BA}$

Forces types

Weight

$\overrightarrow{P} = m\overrightarrow{g}$

Normal

$N$

Tension

$T$

Static Friction

$f_s \leq \mu_sN$

Kinetic Friction

$f_k = \mu_kN$

Elastic Force

$F = -k\Delta x$

Work and Energy

Work

$W = \overrightarrow{F} \cdot \overrightarrow{\Delta x} = |\overrightarrow{F}||\overrightarrow{\Delta x}|cos(\theta)$

Kinetic Energy

$K = \frac{1}{2}mv^2$

Work-Energy Theorem

$W_{neto} = \Delta K$

Power

$P = \frac{dW}{dt} = \overrightarrow{F} \cdot \overrightarrow{v}$

Gravitational Potential Energy

$U_g = mgy$

Elastic Potential Energy

$U_e = \frac{1}{2}kx^2$

Conservative Forces

$\Delta U = -W$

Mechanical Energy

$E = K + U$

Conservatiion of Mechanical Energy

$\Delta E = 0$

Impulse and Linear Momentum

Linear Momentum

$\overrightarrow{p} = m\overrightarrow{v}$

Impulse

$\overrightarrow{I} = \overrightarrow{F}\Delta t$

Impulse-Momentum Theorem

$\overrightarrow{I} = \Delta \overrightarrow{p}$

Conservatiion of Linear Momentum

$\sum \overrightarrow{p_i} = \sum \overrightarrow{p_f}$

Elastic Shocks

$v_{1i} - v_{2i} = -(v_{1f} - v_{2f})$

Shock Inelastic

$K_i \neq K_f$

Gauss's Law: Infinite Cylinder example

• Find the electric field (E) outside a long conducting cylinder with radius R and charge λ per unit length
• Using Gauss's Law: $\oint \overrightarrow{E} \cdot d\overrightarrow{A} = \frac{Q_{enc}}{\epsilon_0}$
• Since $\overrightarrow{E} \parallel d\overrightarrow{A}$ and leveraging cylindrical symmetry: $E(r) \cdot 2\pi r \cdot l = \frac{\lambda l}{\epsilon_0}$
• Result: $E(r) = \frac{\lambda}{2\pi \epsilon_0 r}$

Gauss's Law: Non-Conducting Cylinder

• Find E inside and outside a long non-conducting cylinder (radius $R$, uniform volume charge density $\rho$).

Outside ($r > R$):

• $E(r) \cdot 2\pi r \cdot l = \frac{\rho \cdot \pi R^2 l}{\epsilon_0}$
• $E(r) = \frac{\rho R^2}{2 \epsilon_0 r}$

Inside ($r < R$):

• $E(r) \cdot 2\pi r \cdot l = \frac{\rho \cdot \pi r^2 l}{\epsilon_0}$
• $E(r) = \frac{\rho r}{2 \epsilon_0}$

Gauss's Law: Non-Conducting Sphere

• Find E inside and outside a non-conducting sphere (radius $R$, uniform volume charge density $\rho$).

Outside ($r > R$):

• $E(r) \cdot 4 \pi r^2 = \frac{\rho \cdot \frac{4}{3} \pi R^3}{\epsilon_0}$
• $E(r) = \frac{\rho R^3}{3 \epsilon_0 r^2} = \frac{Q}{4 \pi \epsilon_0 r^2}$

Inside ($r < R$):

• $E(r) \cdot 4 \pi r^2 = \frac{\rho \cdot \frac{4}{3} \pi r^3}{\epsilon_0}$
• $E(r) = \frac{\rho r}{3 \epsilon_0}$

Gauss's Law: Conducting Sphere

• Find E inside and outside a conducting sphere (radius $R$, charge $Q$).

Outside ($r > R$):

• $E(r) = \frac{Q}{4 \pi \epsilon_0 r^2}$

Inside ($r < R$):

• $E(r) = 0$

Gauss's Law: Non-Conducting Plane Sheet

• Find E on both sides of an infinite non-conducting plane sheet (uniform surface charge density $\sigma$).
• Result: $E = \frac{\sigma}{2 \epsilon_0}$

Gauss's Law: Conducting Plate

• Find E inside and outside a conducting plate (thickness $t$, uniform charge density $\rho$).

Inside:

• $E = 0$

Outside:

• $E = \frac{\rho t}{2 \epsilon_0}$ and with $\sigma = \rho t$ we get outside: $E = \frac{\sigma}{\epsilon_0}$