I am trying to learn more about probability mass functions, density functions, expected value, and variance. Are there any online materials or quizzes (with answers and explanations) that I can use to test my understanding?
2026-03-30 21:12:28.1774905148
Any good practice material for expected value and variance?
126 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in PROBABILITY
- How to prove $\lim_{n \rightarrow\infty} e^{-n}\sum_{k=0}^{n}\frac{n^k}{k!} = \frac{1}{2}$?
- Is this a commonly known paradox?
- What's $P(A_1\cap A_2\cap A_3\cap A_4) $?
- Prove or disprove the following inequality
- Another application of the Central Limit Theorem
- Given is $2$ dimensional random variable $(X,Y)$ with table. Determine the correlation between $X$ and $Y$
- A random point $(a,b)$ is uniformly distributed in a unit square $K=[(u,v):0<u<1,0<v<1]$
- proving Kochen-Stone lemma...
- Solution Check. (Probability)
- Interpreting stationary distribution $P_{\infty}(X,V)$ of a random process
Related Questions in STATISTICS
- Given is $2$ dimensional random variable $(X,Y)$ with table. Determine the correlation between $X$ and $Y$
- Statistics based on empirical distribution
- Given $U,V \sim R(0,1)$. Determine covariance between $X = UV$ and $V$
- Fisher information of sufficient statistic
- Solving Equation with Euler's Number
- derive the expectation of exponential function $e^{-\left\Vert \mathbf{x} - V\mathbf{x}+\mathbf{a}\right\Vert^2}$ or its upper bound
- Determine the marginal distributions of $(T_1, T_2)$
- KL divergence between two multivariate Bernoulli distribution
- Given random variables $(T_1,T_2)$. Show that $T_1$ and $T_2$ are independent and exponentially distributed if..
- Probability of tossing marbles,covariance
Related Questions in REFERENCE-REQUEST
- Best book to study Lie group theory
- Alternative definition for characteristic foliation of a surface
- Transition from theory of PDEs to applied analysis and industrial problems and models with PDEs
- Random variables in integrals, how to analyze?
- Abstract Algebra Preparation
- Definition of matrix valued smooth function
- CLT for Martingales
- Almost locality of cubic spline interpolation
- Identify sequences from OEIS or the literature, or find examples of odd integers $n\geq 1$ satisfying these equations related to odd perfect numbers
- property of Lebesgue measure involving small intervals
Related Questions in EXPECTATION
- Prove or disprove the following inequality
- Show that $\mathbb{E}[Xg(Y)|Y] = g(Y) \mathbb{E}[X|Y]$
- Need to find Conditions to get a (sub-)martingale
- Expected Value of drawing 10 tickets
- Martingale conditional expectation
- Variance of the integral of a stochastic process multiplied by a weighting function
- Sum of two martingales
- Discrete martingale stopping time
- Finding statistical data for repeated surveys in a population
- A universal bound on expectation $E[X^ke^{-X}]$
Related Questions in VARIANCE
- Proof that $\mathrm{Var}\bigg(\frac{1}{n} \sum_{i=1}^nY_i\bigg) = \frac{1}{n}\mathrm{Var}(Y_1)$
- $\{ X_{i} \}_{i=1}^{n} \thicksim iid N(\theta, 1)$. What is distribution of $X_{2} - X_{1}$?
- Reason generalized linear model
- Variance of $\mathrm{Proj}_{\mathcal{R}(A^T)}(z)$ for $z \sim \mathcal{N}(0, I_m)$.
- Variance of a set of quaternions?
- Is the usage of unbiased estimator appropriate?
- Stochastic proof variance
- Bit of help gaining intuition about conditional expectation and variance
- Variance of $T_n = \min_i \{ X_i \} + \max_i \{ X_i \}$
- Compute the variance of $S = \sum\limits_{i = 1}^N X_i$, what did I do wrong?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
In my experience teaching intro probability courses, many online instructional materials (even trendy and popular ones) are quirky as to content or hastily written without proofreading. (I have spent many hours over the years dealing with student confusion due to reading poorly written online material.)
I don't want to deprecate online probability instructional materials en masse, but I don't have any suggestions that I can recommend with confidence. If someone else has a recommendation based on personal experience, perhaps it will appear in another Answer.
If you are looking for something that's free online, then I can suggest Grinstead and Snell's 'Intro. to Probability'. It is available in book form, but you can find a free PDF online. Look at Chapter 6.
There are many exercises at the end of the chapter, but I don't know if an answer book is available. However, there are many worked examples. You can choose to view some of them as problems, if you don't peek at the solutions.
Grinstead & Snell is competently written and well-reviewed. It has some examples that are not standard for elementary texts (which was sort of the point of their writing project). Also, the book uses simulation in some places where a traditional text would not. However, there are many very traditional examples, arranged in a reasonable progression of difficulty. I suggest you just skip the examples that are not to your taste. Also, remember that it is Chapter 6, that deals with expectations and variances, so you might want to browse Chapters 1-5 to make sure you have the appropriate background material in mind.