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Randomness and Probability

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Wize University Statistics Textbook > Discrete Random Variables

Randomness and Probability

9 Activities

Let X and Y be discrete random variables where 𝑌=5𝑋+1,𝐸(𝑋)=4𝑌 = 5𝑋 + 1, 𝐸(𝑋) = 4 and 𝑉(𝑋)=2𝑉(𝑋) = 2
More Randomness and Probability Questions:
Randomness and Probability
Let y be the number of cars in a sample of 3 that need to have their cylinder replaced over the next eight years. Here is a probability table that shows the values of y and each individual probability
Find the expected number of cars that will need to have their cylinder replaced
Randomness and Probability
Suppose that certain car parts are manufactured so that their heights are independent of each other, with a mean of μ=31.2\mu=31.2 cm and a standard deviation of σ=0.1\sigma=0.1 cm.
If 2 of these parts are stacked on top of each other to form a new part, what are the mean and standard deviation of this new part?
Randomness and Probability: Expected Value
A certain game wheel consists of 10 equal size slices. 2 of those slices are red, 5 if those slices are blue, and 3 of those slices are green. A player spins the game wheel, and if it lands on red, they lose $10, if it lands on blue they don't win or lose any money, and if it lands on green, they win $6. What is the expected value for a player's winnings?
Randomness and Probability
A certain game wheel consists of 10 equal size slices. 2 of those slices are red, 5 if those slices are blue, and 3 of those slices are green. A player spins the game wheel, and if it lands on red, they lose $10, if it lands on blue they don't win or lose any money, and if it lands on green, they win $6. What is the expected value for a player's winnings?
Randomness and Probability
A biased coin is three times as likely to turn up heads as tails. The coin is tossed twice. Let XX be the number of heads obtained. Which of the following is the probability distribution of XX?
Randomness and Probability
Tony plays a game in which he draws one card. If the card is a heart, he wins $2. Otherwise, he loses $2.
Let X be the amount he will win. Find the standard deviation of X (that is, σx\sigma_x). (round to 3 decimal places)
Randomness and Probability
Let X be a discrete random variable where 𝐸(𝑋+2)=8.𝐸(𝑋 + 2) = 8. Find 𝐸(𝑋)𝐸(𝑋)
Randomness and Probability
Let B be a discrete random variable with 𝐸(𝐵)=6.𝐸(𝐵) = 6. Find 𝐸(−3𝐵−1)𝐸(−3𝐵 − 1)
Randomness and Probability: Independent RV
Let X and Y be two independent random variables where 𝐸(𝑋)=2, 𝐸(𝑌)=5, 𝑉(𝑋)=4, 𝑉(𝑌)=1𝐸(𝑋) = 2, \ 𝐸(𝑌) = 5, \ 𝑉(𝑋) = 4,\ 𝑉(𝑌) = 1.
Randomness and Probability
Suppose that certain car parts are manufactured so that their heights are independent of each other, with a mean of μ=31.2\mu=31.2 cm and a standard deviation of σ=0.1\sigma=0.1 cm.
If 2 of these parts are stacked on top of each other to form a new part, what are the mean and standard deviation of this new part?
Randomness and Probability
Suppose that the random variable XX is the number of complaint calls that a local computer repair store may receive during a day. The probability model of XX is shown below
x01234P(X=x)0.100.100.200.500.10\begin{array}{|c|c|c|c|c|c|} \hline x&0&1&2&3&4\\ \hline P(X=x)&0.10&0.10&0.20&0.50&0.10\\ \hline \end{array}
Given that the expected value of XX is 2.4, what is the standard deviation of XX?
Randomness and Probability
Which of the following are discrete random variables? (Check all that applies.)
Randomness and Probability
Find the expected value and standard deviation.
Randomness and Probability
The exact number of cookies in a box varies from box to box according to the probability distribution function (PDF) given by:
Provide all your answers with at least 2 decimal places (e.g. 0.38)
Randomness and Probability
A bag has three red and four black balls. One ball is drawn at a time, without replacement until a black ball is drawn. Let X represent the number of balls drawn until you get a black one.
Find the probability P(X) of each outcome X. Express P(X) in fractions (e.g. 5/8).
Randomness and probability
A bag has three red and four black balls. One ball is drawn at a time, without replacement until a black ball is drawn. Let X represent the number of balls drawn until you get a black one.
Determine the average number of balls drawn if this experiment were repeated many times. Enter answer in one decimal place X.X (e.g. 2.5)
Probability Density Function
(a) For the following probability density function, the expected value is 36.45. Determine the value of aa.
(b) Find the mean, variance, and standard deviation of the discrete random variable with the given PDF:
Mean:
Randomness and Probability
Amori is good at math and spelling so she entered a math contest and a spelling bee. The
probability of her winning each contest is listed below. Assume the two contests are
independent.
Randomness and Probability
Anna and Louis are newlyweds and plan to have children.
Suppose they have three children. Let X be the number of boys, where
X=0,1,2,3. Complete the probability table:
Probability
Henry is a server at a high end restaurant. X denotes how many times he makes a mistake with an order in a given day. Which of the following is true?
Randomness and probability
You are one of three players remaining in a poker tournament. The total prize is $1,000 where the winner gets $750 and the player who gets second place wins $250. The third-place finisher wins nothing.
Currently, you have the most chips ($$$). Player A has the least amount of chips ($). You are dealt with ace-queen suited. The other two players went all-in with pairs! It is your turn.
If you go on all-in, the pot will be so massive that whoever wins this crucial round will win the tournament. The photo shows that, if you go all-in (or simply call), you have a 27.91% chance of winning this round and knocking both players out - thus winning the tournament! But if you go all-in and lose this round, you will lose a lot of chips and will have no chance in winning the tournament. Instead, you will settle for second or third place, depending if another player gets knocked out or not after this round. If Player A wins this round, all three players remain but Player A will be guaranteed to win the tournament, forcing you and Player B to fight for second place. If Player B wins this round, Player A will be knocked out and you will settle for second place.
Probability
(v) Suppose it costs $10 to play. The prize is $20. What is the expected net profit from a single play?
Discrete vs. Continuous
Which of the following is true?
Probability
(iii) Calculate the expected number of 'Heads' and the standard deviation.