I have two uniform random variables $B$ and $C$ distributed between $(2,3)$ and $(0,1)$ respectively. I need to find the mean of $\sqrt{B^2-4C}$. Could I plug in the means for $B$ and $C$ and then solve or is it more complicated than that? The original question is here: Difference between two real roots with uniformly … Read more
If $X \sim U(-1, 1)$ (so $X$ is uniformly distributed between $-1$ and $1$) and $Y = X^2$, what is the covariance between $X$ and $Y$? Are they independent? So the formula for covariance is: $\operatorname{Cov}(X,Y) = E(XY) – E(X)E(Y)$ Since $Y = X^2$ $E(XY) = E(X^3)$ and $E(Y) = E(X^2)$ and $\operatorname{Cov}(X, Y) = … Read more
Let f(x)=0.025x+0.15 for 2<x<6. The expected value formula is 1/2⋅(b−a). So is the expected value just 1/2⋅(6−2)=4 or do I have to integrate f(x) first? Answer Your distribution is not uniform in [2,6], so the formula 12(b+a) does not hold. Instead, calculate the expected value of X by the general formula as follows E[X]=∫Rxf(x)dx=∫62x(0.025x+0.15)dx=4.1¯3 The … Read more
Suppose uniform[0,1] random variables X_i, i=1,…,\infty are also independent. Whats the following probability? \begin{split} P\Big(\sum_{n=1}^\infty\prod_{i=1}^nX_i<\infty\Big) \end{split} Answer Hint: Show that \mathbb{E} \left( \sum_{n \geq 1} \prod_{i=1}^n X_i \right) = \sum_{n \geq 1} \mathbb{E} \left( \prod_{i=1}^n X_i \right) < \infty. What does this tell you about \mathbb{P} \left( \sum_{n \geq 1} \prod_{i=1}^n X_i < \infty \right) … Read more
I am having troubles with proving almost surely convergence for the following problem: Let $U_j$ be IID $U(0,1)$ distributed and define $A_n$ to be: $A_n=\sum_{k=1}^n \prod_{j=1}^k U_j$ for $n\in \mathbb{N}$. for $n\rightarrow \infty$, I want to prove that $A_n$ converges almost surely to some $A$. I tried with LLN, but this did not give anything. … Read more
Assume customers arrive according to a Poisson process with rate λ=3. Given three customers arrive in the first hour, find the probability that the third customer arrives in the second half of the hour. So that is if X3 is the third arrival time: P(X3∈[0.5,1]∣N(1)=3) and we have X3∼ Gamma(3,3). To solve this I have … Read more
I only know that if it is uniformly distributed the formula for probability function will be 1b−a but in my question I have a uniform distribution on {−1,0,1}. For example if X has uniform distribution on (a,b) then it will be F(x){0x<ax−ab−aa≤x≤b1x>b How do I find uniform distribution on {−1,0,1} ? Answer Isn’t {−1,0,1} a … Read more
In Wikipedia, the formula for the variance of the triangular distribution is given here. However, I don’t know how to find it. I have tried a brute force method but the formula is quite complicated (polynomial of degree 5 in a, b, c) and I can’t simplify it (I tried manually and with Xcas). The … Read more
Here is a question about probability density. I am trying to work it out using a different method from the method on the textbook. But I get a different answer unfortunately. Can anyone help me out? Question: Let X be uniformly distributed random variable in the internal [0,1]. Find the probability density of X2? My … Read more
This question already has answers here: Simulate repeated rolls of a 7-sided die with a 6-sided die (14 answers) Closed 6 years ago. This was a job interview question someone asked about on Reddit. (Link) How can you generate a uniform distribution on $\{1,\dots,7\}$ using a fair die? Presumably, you are to do this by … Read more
