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24 Questions on Random Variable - Assignment | STAT 20, Assignments of Statistics

University of California - BerkeleyStatistics
Prof. E. Anderes

Material Type: Assignment; Professor: Anderes; Class: Introduction to Probability and Statistics; Subject: Statistics; University: University of California - Berkeley; Term: Spring 2007;

Typology: Assignments

Pre 2010

Uploaded on 11/08/2009

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Extra Problems.
More on Random Variables
1. Let Xand Ybe independent copies of a random variable with probability mass
function
f(x) = x2
30,for x= 0,1,2,3,4.
First check that fis indeed a probability mass function for a random variable. Com-
pute sd(X), sd(Y), and sd(X−Y).
2. Toss a fair coin 10 times. Let Yrecord the number of heads in the first 5 tosses and
let Xrecord the number of heads in the last 5 tosses. Find E(X), sd(X), E(Y),
sd(Y), P(X= 2), P(X= 1 and Y≥1).
3. Suppose a box has 7 red tickets, 4 white tickets, and 5 blue tickets. Draw 10 tickets
out of the box with replacement. What is the probability that you draw 3 white or
blue tickets?
4. Suppose I flip a coin 10 times. What is the probability that I get at least 2 tails.
5. Suppose Z∼ N (2,3), Y∼ N (0,4), and Zand Yare independent. Find P(Z > 3),
P(Y > 3), P(Z+Y > 3), and P(Z−Y > 3).
6. Let X1, . . . , X40 be independent copies of a random variable X∼Bin(4, .2). Find
sd(X1+· · · +X40 ) and sd X1+···+X40
40 .
7. Suppose X1, . . . , Xnare nindependent copies a random variable with distribution
N(1,2) and let X=X1+···+Xn
n. Find P(X > 2).
8. Suppose I randomly pick 5 cards with replacement from a standard deck of cards.
What is the probability that three of the cards are diamonds?
9. Suppose X∼Bin(400,0.5). Find a lower bound for P(180 < X < 220).
10. Suppose X∼ N (0,1). Find P(−2< X < 2) and use Chebyshev to find a lower
bound for P(−2< X < 2).
11. Let X1, . . . , X100 be independent copies of a random variable X∼Bin(1, .2). Let
X=X1+···+X100
100 . Use Chebyshev to fill in the following blank. The probability that
Xis between 0.08 and 0.32 is at least .
12. Let X1, . . . , X10000 be independent copies of a random variable X∼Bin(1, .2). Let
X=X1+···+X10000
10000 . Use Chebyshev to fill in the following blank. The probability that
Xis between 0.08 and 0.32 is at least .
pf3

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Extra Problems.

More on Random Variables

  1. Let X and Y be independent copies of a random variable with probability mass function f (x) = x^2 30 , for x = 0, 1 , 2 , 3 , 4.

First check that f is indeed a probability mass function for a random variable. Com- pute sd(X), sd(Y ), and sd(X − Y ).

  1. Toss a fair coin 10 times. Let Y record the number of heads in the first 5 tosses and let X record the number of heads in the last 5 tosses. Find E(X), sd(X), E(Y ), sd(Y ), P (X = 2), P (X = 1 and Y ≥ 1).
  2. Suppose a box has 7 red tickets, 4 white tickets, and 5 blue tickets. Draw 10 tickets out of the box with replacement. What is the probability that you draw 3 white or blue tickets?
  3. Suppose I flip a coin 10 times. What is the probability that I get at least 2 tails.
  4. Suppose Z ∼ N (2, 3), Y ∼ N (0, 4), and Z and Y are independent. Find P (Z > 3), P (Y > 3), P (Z + Y > 3), and P (Z − Y > 3).
  5. Let X 1 ,... , X 40 be independent copies of a random variable X ∼ Bin(4, .2). Find sd(X 1 + · · · + X 40 ) and sd

( X 1 +···+X 40

40

  1. Suppose X 1 ,... , Xn are n independent copies a random variable with distribution N (1, 2) and let X = X^1 +··· n +Xn. Find P (X > 2).
  2. Suppose I randomly pick 5 cards with replacement from a standard deck of cards. What is the probability that three of the cards are diamonds?
  3. Suppose X ∼ Bin(400, 0 .5). Find a lower bound for P (180 < X < 220).
  4. Suppose X ∼ N (0, 1). Find P (− 2 < X < 2) and use Chebyshev to find a lower bound for P (− 2 < X < 2).
  5. Let X 1 ,... , X 100 be independent copies of a random variable X ∼ Bin(1, .2). Let X = X^1 +··· 100 +X 100. Use Chebyshev to fill in the following blank. The probability that X is between 0.08 and 0.32 is at least.
  6. Let X 1 ,... , X 10000 be independent copies of a random variable X ∼ Bin(1, .2). Let X = X^1 +··· 10000 +X^10000. Use Chebyshev to fill in the following blank. The probability that X is between 0.08 and 0.32 is at least.
  1. Suppose a friend gives you a coin and tells you that it is either a fair coin or an unfair coin that lands heads with probability 0.2. You decided to flip the coin 100 times, compute the proportion of heads, then guess that the coin is the unfair one if the proportion of heads is between 0.08 and 0.32. If the coin your testing is really the unfair one, use Chebyshev to get a lower bound on the probability that you guess correctly.
  2. Suppose I have a box with 10 tickets. Each ticket has a number. There are three 1’s, two 11’s and five 12’s. Now draw n tickets at random with replacement and let Y 1 ,... , Yn denote the resulting numbers. Find sd(Y 1 + · · · + Yn) and sd( Y^1 +··· n +Yn).
  3. Suppose I have a box with 10 tickets. Each ticket has a number. There are three 1’s, two 11’s and five 12’s. Now draw n tickets at random with replacement and let X count the number of 11’s in the n draws. Find P (X = 2) and sd(X). Also find the standard deviation of the proportion of 11’s in the n draws.
  4. Explain in each case why the given equation cannot serve as the probability density function of a random variable that takes on values on the interval from 1 to 4.

(a) f (x) = 0.25; (b) f (x) = 19 (4x − 7).

  1. Suppose that a random variable has the following density

f (x) =

1 / 30 for 20 < x < 50 0 otherwise.

Find the probabilities that this random variable will take on a value

(a) from 30 to 50; (b) less than 20; (c) greater than 45; (d) between 25 and 35.

  1. The probabilities that a person convicted of drunk driving will spend a night in jail, have his license revoked, or both are, respectively, 0.68, 0.51, and 0.22. What is the probability that a person convicted of drunk driving will spend a night in jail and/or have his license revoked.
  2. Let Z ∼ N (0, 1). Find the number z such that P (Z > z) = 0.5.
  3. Let Y ∼ N (0, 1). Find the number w such that P (|Y | > w) = 0.5, where |Y | denotes the absolute value of Y ).