Probability and Statistics: Discrete Random Variables
What are the probabilities of collecting six frogs, determining their sexes, and getting a result in which there are zero males, one male, two males, etc., from those six frogs?
What are the probabilities of collecting six frogs, determining their sexes and getting a result in which there are zero males, one male, two males, etc., from those six frogs?
Define a random variable capital X where it is equal to 1 if heads and 0 if tails when flipping a fair coin.
Define another random variable capital Y as the sum of the upward face after rolling 7 dice.
Hugo plans to buy packs of baseball cards until he gets the card of his favorite player, but he only has enough money to buy at most four packs. Suppose that each pack has a probability of 0.2 of containing the card that Hugo is hoping for. Let the random variable X be the number of packs of cards Hugo buys. Find the indicated probability: what is the probability that X is greater than or equal to two?
Calculate the expected value of winning a game where if she wins, she receives 100. The probability of winning the game is 20%.
Company XYZ generates a profit of 500 for each defective laptop. If 3 out of every 100 laptops produced are defective, calculate the expected value of profit per laptop.
Given the probability distribution for the random variable x, representing the number of workouts in a week, find the expected value (mean) of x.
Determine the PMF of a random variable which is a function of two other random variables X and Y, by finding the probability that the function of X and Y takes on a specific numerical value.
Coming home from work, you always seem to hit every single light. Throughout time, you calculate the odds of making it through any light to be 0.2. On any given night, how many lights can you expect to hit before finally making it through one, and with what standard deviation? Finally, what's the probability of the third light you come across being the first one that is green?
A game is played where a player rolls a fair six-sided die until a 1 is rolled. What is the expected mean and standard deviation of the number of rolls?
Instead of a six-sided die, assume a 12-sided die is used in the game. Calculate the mean and standard deviation of the number of rolls until a 1 is rolled.
Calculate the mean and variance of a linear combination of independent random variables given their means and variances.
Discuss the concept of a probability mass function (PMF) for a discrete variable using a six-sided dice example, where each outcome has a probability of rac{1}{6}.
Explain the cumulative distribution function (CDF) for the given discrete variable example and how it relates to cumulative probability.
Describe the concept of a rigged six-sided dice where threes and fours cannot be rolled, affecting the probability mass function (PMF) and cumulative distribution function (CDF).