🎲 probability

1. The problem asks for the possible values of the random variable $T$, which represents the number of tails when two coins are tossed. 2. When tossing two coins, each coin can be

1. **Problem statement:** Given probabilities in a Venn diagram with events A, B, C, D and some intersections, find mutually exclusive events, values of unknown probabilities, cond

1. **Problem statement:** We are given a joint probability density function (pdf) for random variables $X$ and $Y$:

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

1. **Problem Statement:** Given the joint pdf of continuous random variables $X$ and $Y$: $$f(x,y) = \frac{1}{8}(x+y), \quad 0 < x < 2, 0 < y < 2; \quad 0 \text{ otherwise}$$

1. **Problem Statement:** Calculate the covariance $\sigma_{xy}$ and correlation coefficient $\rho_{xy}$ for discrete random variables $X$ and $Y$ given their joint PMF.

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