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1. Consider a statistical experiment E: (, F,P) and an event A . Note: A EF....
Let 2 [0, 1], and let F be the collection of every subset of such that the subset or its compleme...
Let 2 [0, 1], and let F be the collection of every subset of such that the subset or its complement is countable. Let P(.) be a measure on F such that for A E F, P(A) if A is countable and P(A)1 if Ac is countable. (a) Is F a field? Also, is F a σ-field? (Note that afield is closed under finite union while a σ-field is closed under countable union. (b) Is P finitely additive? Also, is...
Consider an experiment corresponding to the single throw of a die. Let E1 be the event...
Consider an experiment corresponding to the single throw of a die. Let E1 be the event that the throw is one of 11,2,3,4) and let E2 the event that the throw is one of 13,4,5,6). You are given that P(E) P(E2) 2/3. (a) Identify the smallest o-algebra sf containing the events E1 and E2. (b) Attach probabilities P(J) to all the elements J E (c) What can you say about the probability P(11]) of the event associated to a throw...
1. If two events are independent how do we calculate the and probability, P(E and F),...
1. If two events are independent how do we calculate the and probability, P(E and F), of the two events? (As a side note: this "and" probability, P(E and F), is called the joint probability of Events E and F. Likewise, the probability of an individual event, like P(E), is called the marginal probability of Event E.) 2. One way to interpret conditional probability is that the sample space for the conditional probability is the "conditioning" event. If Event A...
Consider a binomial experiment with n-12 and p 0.2. a. Compute f (0) (to 4 decimals)....
Consider a binomial experiment with n-12 and p 0.2. a. Compute f (0) (to 4 decimals). f(0) b. Compute f(8) (to 4 decimals) f(8) C. Compute P(z 〈 2) (to 4 decimals). d. Compute P(x 2 1) (to 4 decimals). P( 21) e. Compute E(x) (to 1 decimal) E(x) f. Compute Var(z) and σ. Var(x) - (to 2 decimals) (to 2 decimals)
Consider a binomial experiment with n- 12 and p0.2 a. Compute f(0) (to 4 decimals). f(0)...
Consider a binomial experiment with n- 12 and p0.2 a. Compute f(0) (to 4 decimals). f(0) b. Compute f (8) (to 4 decimals). f(8) c. Compute P(x < 2) (to 4 decimals) Pa 2) d. Compute P1 (to 4 decimals). e. Compute E(z) (to 1 decimal). E(x) f. Compute Var(z) and σ. Var(x) (to 2 decimals) to 2 decimals) f. Compute the probability of six occurrences in three time periods (to 4 decimals).
Part III – Probability and Statistics Each question is worth 4 points. 1. Consider the following...
Part III – Probability and Statistics Each question is worth 4 points. 1. Consider the following experiment and events: two fair coins are tossed, E is the event "the coins match”, and F is the event “at least one coin is Heads”. (a) Find the probabilities P(E), P(F), P(EUF), and P(En F). (b) Are the events and F independent? Explain. 2. Let X be a discrete random variable with the probability function given by f(2) k(x2 – 2x) + 0.2...
Problem 1-5 1. If X has distribution function F, what is the distribution function of e*?...
Problem 1-5
1. If X has distribution function F, what is the distribution function of e*? 2. What is the density function of eX in terms of the densitv function of X? 3. For a nonnegative integer-valued random variable X show that 4. A heads or two consecutive tails occur. Find the expected number of flips. coin comes up heads with probability p. It is flipped until two consecutive 5. Suppose that PX- a p, P X b 1-p, a...
If E C F, then show that (a) Fe C Ee ss (b) P(Fe) P(Ee). Use...
If E C F, then show that (a) Fe C Ee ss (b) P(Fe) P(Ee). Use the probability axioms and other identities in your proof. where the inequality holds by P(FE) 2 0
1. Consider the experiment: You flip a coin once and roll a six-sided die once. Let...
1. Consider the experiment: You flip a coin once and roll a six-sided die once. Let A be the event that you roll an even number and B be the event that you flip heads. (a) Determine the sample space S for this experiment. (Hint: There are 12 elements of the sample space.) (b) Which outcomes are in A? (c) Which outcomes are in B? (d) Which outcomes are in A'? What does it mean in words? (e) Which outcomes...
Assume that events (E, F) are disjoint, and their probabilities are specified as (here p. An...
Assume that events (E, F) are disjoint, and their probabilities are specified as (here p. An experiment is repeated until either E or F will occur Find the probability that E will occur before F Hint Introduce a random variable, N, which is the first occurrence of EUF. Then express the probability that E occurs before F, given that EUF occurs at the time N and use the formula where A is the desired event
Let 2 [0, 1], and let F be the collection of every subset of such that the subset or its complement is countable. Let P(.) be a measure on F such that for A E F, P(A) if A is countable and P(A)1 if Ac is countable. (a) Is F a field? Also, is F a σ-field? (Note that afield is closed under finite union while a σ-field is closed under countable union. (b) Is P finitely additive? Also, is...
Consider an experiment corresponding to the single throw of a die. Let E1 be the event that the throw is one of 11,2,3,4) and let E2 the event that the throw is one of 13,4,5,6). You are given that P(E) P(E2) 2/3. (a) Identify the smallest o-algebra sf containing the events E1 and E2. (b) Attach probabilities P(J) to all the elements J E (c) What can you say about the probability P(11]) of the event associated to a throw...
Consider a binomial experiment with n-12 and p 0.2. a. Compute f (0) (to 4 decimals). f(0) b. Compute f(8) (to 4 decimals) f(8) C. Compute P(z 〈 2) (to 4 decimals). d. Compute P(x 2 1) (to 4 decimals). P( 21) e. Compute E(x) (to 1 decimal) E(x) f. Compute Var(z) and σ. Var(x) - (to 2 decimals) (to 2 decimals)
Consider a binomial experiment with n- 12 and p0.2 a. Compute f(0) (to 4 decimals). f(0) b. Compute f (8) (to 4 decimals). f(8) c. Compute P(x < 2) (to 4 decimals) Pa 2) d. Compute P1 (to 4 decimals). e. Compute E(z) (to 1 decimal). E(x) f. Compute Var(z) and σ. Var(x) (to 2 decimals) to 2 decimals) f. Compute the probability of six occurrences in three time periods (to 4 decimals).
Part III – Probability and Statistics Each question is worth 4 points. 1. Consider the following experiment and events: two fair coins are tossed, E is the event "the coins match”, and F is the event “at least one coin is Heads”. (a) Find the probabilities P(E), P(F), P(EUF), and P(En F). (b) Are the events and F independent? Explain. 2. Let X be a discrete random variable with the probability function given by f(2) k(x2 – 2x) + 0.2...
Problem 1-5
1. If X has distribution function F, what is the distribution function of e*? 2. What is the density function of eX in terms of the densitv function of X? 3. For a nonnegative integer-valued random variable X show that 4. A heads or two consecutive tails occur. Find the expected number of flips. coin comes up heads with probability p. It is flipped until two consecutive 5. Suppose that PX- a p, P X b 1-p, a...
If E C F, then show that (a) Fe C Ee ss (b) P(Fe) P(Ee). Use the probability axioms and other identities in your proof. where the inequality holds by P(FE) 2 0
1. Consider the experiment: You flip a coin once and roll a six-sided die once. Let A be the event that you roll an even number and B be the event that you flip heads. (a) Determine the sample space S for this experiment. (Hint: There are 12 elements of the sample space.) (b) Which outcomes are in A? (c) Which outcomes are in B? (d) Which outcomes are in A'? What does it mean in words? (e) Which outcomes...
Assume that events (E, F) are disjoint, and their probabilities are specified as (here p. An experiment is repeated until either E or F will occur Find the probability that E will occur before F Hint Introduce a random variable, N, which is the first occurrence of EUF. Then express the probability that E occurs before F, given that EUF occurs at the time N and use the formula where A is the desired event
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