1
vote
2answers
31 views

Approximating a probability from a sample of size $200$

I'm having some issues with this problem. The median age of residents of US people is $35.6$ years. If a survey of $200$ residents is taken, approximate the probability that at least $110$ will be ...
0
votes
1answer
30 views

weighted sum of exponential random variables

Suppose $X_i$ are i.i.d random variables and $X\sim\operatorname{Exp}(\lambda)$ i.e. $Pr(X\le x)=1-e^{-\lambda x}$ for $x \ge 0$. What is the density function of $Z=\sum_{i=1}^{N}\alpha_iX_i$ where ...
0
votes
0answers
14 views

Exponential Order Statistics Independence

Are the order statistics from the $n$-sample with $X_i\sim \text{Exp}(\lambda)$ (taking, without loss of generality, $\lambda=1$) $\Delta_{(k)}X=X_{(k)}-X_{(k-1)}$ independent? Can show that for an ...
1
vote
0answers
19 views

Calculate Expectancy of Poisson distribution with Poisson parameter

The problem: Let X have the Poisson distribution with parameter $\lambda$, where $\lambda >0$. Calculate E[X!] for every possible value of $\lambda$, So what I got so far: $$E[X!] = ...
-1
votes
1answer
24 views

Proof for one of the Cumulative Distribution Function

Hey I tried searching online but I can't find this proof based on the CDF. $$P(a < X \le b) = F(b) - F(a)$$ Thanks for any help.
0
votes
1answer
30 views

deriving the distribution of Y having information about X

A child who has swallowed a pin of length 4cm is X-rayed. THe pin appears on the X-ray film as shown below, the length of the image being y, an observation on the random variable Y. The pin is at an ...
0
votes
2answers
36 views

Conditional Expectation of Exponential Order Statistic $\text{E}(X_{(2)} \mid X_{(1)}=r_1)$

Having already worked out the distributions of $\Delta_{(2)}X=X_{(2)}-X_{(1)}\sim\text{Exp}(\lambda)$ and of $\Delta_{(1)}X=X_{(1)}\sim\text{Exp}(2\lambda)$ where $X_{(i)}$ are the $i$th order ...
2
votes
2answers
32 views

Clarification on expected number of coin tosses for specific outcomes.

As seen in this question, André Nicolas provides a solution for 5 heads in a row. Basically, for any sort of problem that relies on determining this sort of probability, if the chance of each event ...
2
votes
2answers
43 views

Understanding $P(X=n)=0$ when $X$ follows a continuous distribution

I have something that I can't get about continuous distribution. Let's say a variable $X$ follows a continuous distribution on $]0;1[$. Then when I pick a completely random number on $]0;1[$, say I ...
0
votes
0answers
24 views

How to compute $p(\pi>\theta_2+N>\theta_1+\pi, \hspace{1em}\theta_1<0, \hspace{1em}\theta_2<\theta_1+\pi)$ for dependent jointly Normal RVs

How would one go about computing: $p(\pi>\theta_2+N>\theta_1+\pi, \hspace{1em}\theta_1<0, \hspace{1em}\theta_2<\theta_1+\pi)$ given $\theta_1, \theta_2$ are jointly Normal (mean=0, ...
3
votes
1answer
61 views

Exponential Distribution Function

If $X\sim \text{Exp}(X)$ then for all positive $a$ and $b$, $P(X>a+b\mid X>a)=P(X>b).$ So given independent random variables $X \sim \text{Exp}(\lambda)$, $Y \sim\text{Exp}(\mu)$ we would ...
0
votes
1answer
60 views

Probability distribution question

Suppose that X is Poisson distributed with mean $\lambda >0$ and Y is geometrically distributed with parameter $p\in(0, 1)$. Assume that X and Y are mutually independent. How do I show that $P(Y ...
0
votes
0answers
19 views

Gap distribution independence proof

I have a question bout the proof of the independence of gap RVs. Given the independent exponentially distributed random variables $\xi_1$, $\xi_2$ ~ $\text{Exp}(\lambda)$, and a corresponding order ...
0
votes
0answers
51 views

Find density for $Z=X+Y$ with joint density function

Find the density function of $Z=X+Y$, $X,Y$ have the joint density function $f(x,y) = \frac{1}{2} (x+y) e^{-(x+y)},\, x,y \geq 0$. My initial idea is to calculate the distribution function of Z like ...
0
votes
1answer
24 views

Moment generating function using probability function?

Suppose that $X_1, X_2, ..., X_n$ are independent, where each $X_i$ has probability (mass) function $p_i(x_i)$ given as $p_i(x_i) = \frac{e^{-\lambda} \lambda_i^{x_i}}{x_i!}$ (only the parameter ...

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