Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [discrete-distributions]

79 questions
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Let $X,X_1,X_2,X_3,...$ be positive integer random variables. Show that $X_n \overset{d}{\to} X$ implies $\lim_{n\to\infty} P(X_n=k) = P(X=k)$

Question Let $X,X_1,X_2,X_3,...$ be positive integer random variables. Show that $X_n \overset{d}{\to} X$ implies $\lim_{n\to\infty} P(X_n=k) = P(X=k)$. The $\overset{d}{\to}$ denotes convergence in distribution. Attempt Here I try to show $$X_n…
EssentialAnonymity
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How to interpret sum of two random variables that cross domains?

suppose we have two discrete random variables: $X: \{$6 sided dice rolls$\}$ $\rightarrow \{1..6\}$ (following uniform distribution) $Y: \{$coin flips$\}$ $\rightarrow \{0,1\}$ (following uniform distribution) How can we add X+Y=Z, for Z a random…
user352102
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If $X$ and $Y$ have the same marginal distribution, then do they have to have the same conditional distribution?

Suppose $X$ and $Y$ are two random variables that have the same distribution. Does $$P[X \leq t \mid Y=a]$$ be necessarily equal to $$\;\ P[Y \leq t \mid X=a]?$$ Note that if $X$ and $Y$ are bivariate normal with correlation $\rho$ and each is…
John L
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distribution bootstrap sample median

I am interested in the conditional probability that the median $X^*_{(m)}$ of a bootstrap sample $X_1^*,\ldots,X_n^*$, where $n=2m-1$ for integer $m$, equals the $k$th order statistic $X_{(k)}$ of the original sample…
Christoph Hanck
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Finding the mode given the probability of occurence

When a teacher asks a question, a student has a probability of 0.4 of being asked. Assume the occurrence is independent. What is the mode of the number of questions raised by the teacher it takes for the same student to be asked 2 questions? I am…
user672518
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Sufficient Statistics and Discrete Distributions

Let $X_1, \ldots, X_n$ be a random sample of size $n$ from the following distribution: $$f(x;\theta) = \left\{\begin{array}{ccc} \frac{1 - \theta}{6} & , & x = 1 \\ \frac{1 + \theta}{6} & , & x = 2 \\ \frac{2 - \theta}{6} & , & x = 3 \\ \frac{2 +…
Chesso
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Distribution of X+U when X is a discrete and U is a continous random variable

Suppose $X$ and $U$ are independent random variables. $X$ is a discrete uniform variable and $U$ is a continuous uniform $[0,1]$ variable. What is the value of $\mathbb P(X+U\leq y)$, where $y$ is a real number?
Nisha
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The meaning of a parameterization of the logarithmic distribution

In calculus one learns that $$ p + \frac{p^2} 2 + \frac{p^3} 3 + \frac{p^4} 4 + \cdots = -\log(1-p). \tag 1 $$ Thus a discrete probability distribution on the set $\{1,2,3,\ldots\}$ is given by $$ \Pr(X=x) = \frac{-p^x}{ x\log(1-p)} \text{ for } x =…
Michael Hardy
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What is the distribution of the difference of two independent multinomial random variables?

Say I have two independent random vectors $X_c$ and $X_f$. The random vector $X_c$ is composed by three random variables: $X_{1c}$, $X_{2c}$ and $X_{3c}$. The second random vector $X_f$ is composed by $X_{1f}$, $X_{2f}$ and…
Giovani Paludo
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Kernel Density Estimation for a Discrete Variable

I was tying to estimate the distribution for a discrete variable. However, suddenly I thought that "Is a simple histogram sufficient? because I have observations for every evaluation point" So, my question is that "Is there any reason we have to use…
QWEQWE
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Expression for Probability of Being Between Two Poisson Random Variables?

I have two independent Poisson random variables $A \sim \text{Poisson}(\lambda_A)$ and $B \sim \text{Poisson}(\lambda_B)$. For a fixed given integer $k$, I'd like to determine $$P(A < k \leq A + B).$$ Is there an analytical expression for this…
Rylan Schaeffer
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Can we always write a random variable as conditional expectation plus error?

Consider the random variables $Y,X$. I believe that we can always write $$ Y=E(Y|X)+\epsilon $$ with $E(\epsilon|X)=0$. Question: Is the above true regardless whether $Y$ is a discrete or continuous random variable? My thoughts: I believe that the…
TEX
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Conditional expectaction with probabilities for a sum of independent random variable

I have a r.v $S_N$ built as a sum of Bernoulli with parameter $p$. So $S_N = X_1 + X_2 + \ldots + X_N$. There is a second variable N, such that $N \sim Poisson(\lambda) $. I have to compute: $P(S_N=0)$ $\mathop{\mathbb{E}}(S_N \ | \ N = 4…
docdev
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Do financial return series have a probability mass function (pmf)?

Stock returns, computed from stock prices as $r_t = \ln (p_{t}) - \ln (p_{t-1})$, are real-valued and unbounded giving the impression that they are continuous random variables. But aren't they actually discrete random variables given…
develarist
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Sampling distribution of the mean of the discrete-power law distribution

For a certain problem I wish to generate random integers $k$ so that their distribution follows $p_k \sim k^{-\alpha}$ for $k \geq k_{\text{min}}$, $k_{\text{min}} > 0$. I am following the procedure given in this review (page 699). Now the problem…
Peaceful
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