STAT 400 Spring 2015 Discussion 3 1 – 2. Sally sells seashells by the seashore. The daily sales X of the seashells have the following probability distribution: x f ( x ) 0 1 2 3 4 0.25 0.20 0.15 0.10 1. a) Find the missing probability f ( 0 ) = P ( X = 0 ). b) Find the probability that she sells at least three seashells on a given day. c) Find the expected daily sales of seashells. d) Find the standard deviation of daily sales of seashells. 2. Suppose each shell sells for $5.00. However, Sally must pay $3.00 daily for the permit to sell shells. Let Y denote Sally’s daily profit. Then Y = 5 ⋅ X – 3. e) Find the probability that Sally’s daily profit will be at least $10.00. [ “Hint”: How many shells must Sally sell to have daily profit of at least $10.00? ] f) Find Sally’s expected daily profit. g) Find the standard deviation of Sally’s daily profit.3. From a group of 16 male and 9 female armadillos, Noah must choose two to travel on his ark. Unable to distinguish between male and female armadillos, Noah must choose at random. a) Noah chooses the two armadillos at random. Compute the probability that Noah gets two armadillos of the opposite sex (i.e., one male and one female armadillo). b) In order to improve his chances of selecting at least one male and one female armadillo, Noah decides to "cheat" and select three armadillos to travel on his ark. Compute the probability that Noah gets at least one male and one female armadillo. 4. a) How many ways are there to rearrange letters in the word STATISTICS? b) If three letters are selected at random from the word STATISTICS, what is the probability that the selected letters can be rearranged to spell CAT? ( "Hint": The order of the selection is NOT important, the selection is done without replacement. ) That is, what is the probability that selected letters are A, C, and T? c) If six letters are selected at random from the word STATISTICS, what is the probability that the selected letters can be rearranged to spell ASSIST?5. A box contains 2 green and 3 red marbles. A person draws a marble from the box at random without replacement. If the marble drawn is red, the game stops. If it is green, the person draws again until the red marble is drawn (note that total number of marbles drawn cannot exceed 3 since there are only 2 green marbles in the box at the start). Let the random variable X denote the number of green marbles drawn. a) Find the probability distribution of X. [ Hint: There are only 3 possible outcomes for this experiment. It would be helpful to list these three outcomes ( three possible sequences of colors of marbles drawn ). It may be helpful to make a tree diagram for this experiment. Remember that all ( three ) probabilities must add up to one. ] b) Suppose it costs $5 to play the game, and the person gets $10 for each green marble drawn. Is the game fair * ? Explain. [ Hint: Find µ X = E(X), the average (expected) number of green marbles drawn per game. ]6. Suppose a discrete random variable X has the following probability distribution: P ( X = 1 ) = ln 3 – 1, P ( X = k ) = ()! 3lnkk, k = 2, 3, … Recall ( Discussion #1 ): this a valid probability distribution. Find µ X = E ( X ). 7. Let X be a discrete random variable with p.m.f. p ( x ) that is positive on the even non-negative integers { 0, 2, 4, 6, 8, … } and is zero elsewhere. Suppose p ( 0 ) = 87, p ( k ) = k 31, k = 2, 4, 6, 8, … . Recall ( Discussion #1 ): this a valid probability distribution. Find µ X = E ( X ). 8. Suppose a discrete random variable X has the following probability distribution: f ( k ) = k 5100, k = 3, 4, 5, 6, … . Recall ( Discussion #1 ): this a valid probability distribution. Find µ X = E ( X ).1 – 2. Sally sells seashells by the seashore. The daily sales X of the seashells have the following probability distribution: x f (x) x ⋅ f (x) ( x − µ ) 2 ⋅ f (x) x 2 ⋅ f (x) 0 1 2 3 4 0.30 0.25 0.20 0.15 0.10 0.00 0.25 0.40 0.45 0.40 0.6750 0.0625 0.0500 0.3375 0.6250 0.00 0.25 0.80 1.35 1.60 1.00 1.50 1.7500 4.00 1. a) Find the missing probability f ( 0 ) = P ( X = 0 ). f ( 0 ) = 1 − [0.25 + 0.20 + 0.15 + 0.10] = 1 − 0.70 = 0.30. b) Find the probability that she sells at least three seashells on a given day. P(X ≥ 3) = P(X = 3) + P(X = 4) = 0.15 + 0.10 = 0.25. c) Find the expected daily sales of seashells. E(X) = µ X = ∑⋅xxfx all)( = 1.50.d) Find the standard deviation of daily sales of seashells. ( )∑⋅−==xxfx all22 X)( Var(X) μσ = 1.75. OR [ ]2 all22 X E(X) )( Var(X)σ −==∑⋅xxfx = 4.00 − (1.50) 2 = 4.00 − 2.25 = 1.75. σ X = SD(X) = 75.1 = 1.3229. 2. Suppose each shell sells for $5.00. However, Sally must pay $3.00 daily for the permit to sell shells. Let Y denote Sally’s daily profit. Then Y = 5 ⋅ X – 3. e) Find the probability that Sally’s daily profit will be at least $10.00. [ “Hint”: How many shells must Sally sell to have daily profit of at least $10.00? ] x y f (x) = f (y) 0 1 2 3 4 −$3.00 $2.00 $7.00 $12.00 $17.00 0.30 0.25 0.20 0.15 0.10 1.00 P(Y ≥ 10) = P(X ≥ 3) = 0.25. ( Sally’s daily profit is at least $10.00 if and only if she sells at least 3 shells. ) f) Find Sally’s expected daily profit. µ Y = E(Y) = $5.00 ⋅ E(X) − $3.00 = $5.00 ⋅ 1.50 − $3.00 = $4.50. ( On average, Sally sells 1.5 shells per day, her expected revenue is $7.50. Her expected profit is $4.50 since she has to pay $3.00 for the permit. )OR x y f (x) = f (y) y ⋅ f (y) 0 1 2 3 4 −$3.00 $2.00 $7.00 $12.00 $17.00 0.30 0.25 0.20 0.15 0.10 −0.90 0.50 1.40 1.80 1.70 1.00 4.50 µ Y = E(Y) = ∑⋅yyfy all)( = $4.50. g) Find the standard deviation of Sally’s daily profit. σ Y = SD(Y) = 5 ⋅ SD(X) = 5 ⋅ 1.3229 = $6.6145. OR x y f (x) = f (y) ( y − µ Y ) 2 ⋅ f (y) y 2 ⋅ f (y) 0 1 2 3 4 −$3.00 $2.00 $7.00 $12.00 $17.00 0.30 0.25 0.20 0.15 0.10 …
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