🎲 probability

1. **Problem Statement:** Given the joint density function $$f(x,y) = x + y$$ for $$0 < x \leq 1$$ and $$0 < y < 1$$, and zero elsewhere, we need to find: (i) $$E(X)$$

1. **Problem Statement:** Calculate the covariance $\mathrm{Cov}(X,Y)$ given the joint probability distribution table: | X \ Y | 0 | 1 | 2 |

1. **Problem Statement:** Given the joint pmf of discrete random variables $X$ and $Y$:

1. **Problem Statement:** We have a joint density function for random variables $Y_1$ and $Y_2$ given by:

1. **Problem statement:** Given the joint probability density function (pdf) $$f(x,y) = \begin{cases} \frac{3}{2} y^2 & 0 < x < 2, 0 < y < 1 \\ 0 & \text{otherwise} \end{cases}$$ F

1. **Problem 9(a): Find the marginal probability function of X** given joint probability function $$f(x,y) = \frac{1}{54}(3x + 2y - 4)$$ for $$x=1,2,3$$ and $$y=1,2,3$$. 2. The mar

1. **Problem statement:** Given the joint probability function $$f(x, y) = \frac{1}{54} (3x + 2y - 4)$$

1. **State the problem:** We have a random variable $X$ with probability density function (PDF) $$ f(x) = \begin{cases} 1 + x, & -1 < x \leq 0 \\ 1 - x, & 0 < x < 1 \\ 0, & \text{o

1. **Problem Statement:** We have two fair spinners with numbers on their sectors. The first spinner has 4 sectors labeled 6, 3, 4, 5. The second spinner has 3 sectors labeled 3, 4

1. **Problem statement:** Ravi sells an average of 3 life insurance policies per week. We use Poisson distribution to find probabilities for different scenarios. 2. **Poisson distr

1. **State the problem:** We have a probability distribution with values of $k$ and their probabilities $P(k)$:

1. **State the problem:** We want to find the probability that the first digit $D$ in a data entry is greater than or equal to 2, i.e., $P(D \geq 2)$. 2. **Recall the probability d

1. **State the problem:** We need to find the mean (expected value) of the random variable $X$, which represents the number of packs Rodrigo buys until he gets his favorite card. 2

1. **Problem statement:** Given the joint probability density function (pdf) of random variables $X$ and $Y$: $$f_{X,Y}(x,y) = \frac{2}{3}(x + xy), \quad 0 < x < 1, 0 < y < 2; \qua

1. **Problem statement:** Given the joint probability function $$f(x,y) = \frac{1}{54}(3x + 2y - 4)$$ for $$x = 1, 2, 3$$ and $$y = 1, 2, 3$$, find the marginal probability functio

1. **Problem Statement:** Given the joint probability mass function (pmf) of random variables $X$ and $Y$:

1. **Problem Statement:** Given the joint probability density function (pdf) for $(X_1, X_2)$ with $-1 \leq \alpha \leq 1$:

1. **Problem Statement:** Given the joint pdf $$f(x,y) = \frac{2}{(1+x+y)^3}$$ for $$x>0, y>0$$ and 0 elsewhere, find: (i) Marginal densities of $$X$$ and $$Y$$

1. **Problem statement:** Given the joint probability density function (pdf) of $X$ and $Y$: $$p(x,y) = \begin{cases} k(8 - x - y), & 0 < x < 3, 0 < y < 4 \\ 0, & \text{otherwise}

1. **Problem Statement:** We have a joint probability density function (p.d.f) of random variables $X$ and $Y$ given by:

1. **Problem Statement:** (a) Given joint pdf $f_{X,Y}(x,y) = \frac{15}{2}(2 - x - y)$ for $0 < x < 1$, $0 < y < 1$, find the conditional density $f_{X|Y}(x|y)$.