X is a negative binomial random variable with parameters. r=1 and P(S)=p p=62/100. Show that the...
X is a negative binomial random variable with parameters. r=1 and P(S)=p p=62/100. Show that the probability mass function for x is well defined. That it satisfies the requirement for any discrete pmf
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X is a negative binomial random variable with parameters. r=1
and P(S)=p p=62/100. Show that the...
If X is a nonnegative integer-valued random variable then the function P(z), defined for lzl s...
If X is a nonnegative integer-valued random variable then the function P(z), defined for lzl s 1 by is called the probability generating function of X (a) Show that d* (b) With 0 being considered even, show that PX is even) = P(-1) + P(1) (c) If X is binomial with parameters n and p, show that Pix is even) -1+12p (d) If X is Poisson with mean A, show that 1+ e-24 2 P[X is even)- (e) If X...
Let X be random variable with the binomial distribution with parameters n and 0 < p < 1. (1) Show that (P(X = x) /...
Let X be random variable with the binomial distribution with parameters n and 0 < p < 1. (1) Show that (P(X = x) / P(X = x -1)) - 1 = np + (p - x)) / (x(1-p)) for any 1 ≤ x ≤ n. (2) Show that when 0 ≤ x < (n + 1)p , P(X = x) is an increasing function x and for (n + 1)p < x ≤ n, P(X = x) is a...
The moment generating function ф(t) of random variable X is defined for all values of t by et*p(x), if X is discrete e...
The moment generating function ф(t) of random variable X is defined for all values of t by et*p(x), if X is discrete e f (x)dx, if X is continus (a) Find the moment generating function of a Binomial random variable X with parameters n (the total number of trials) and p (the probability of success). (b) If X and Y are independent Binomial random variables with parameters (n1 p) and (n2, p), respectively, then what is the distribution of X...
Problem 7 (15 points). Let X be random variable with the binomial distribution with parameters n...
Problem 7 (15 points). Let X be random variable with the binomial distribution with parameters n and 0 <p<1. (1) Show that **- 1 = 2* for any 1 Sxsn. (2) Show that when 0 < x < (n + 1)p, P(X = x) is an increasing function x and for (n + 1)p <x Sn, P(X = x) is a decreasing function x. (3) A certain basketball player makes a foul shot with probability 0.80. Determine for whal value...
3. You are given a binomial random variable X, with parameters n = 8 and p...
3. You are given a binomial random variable X, with parameters n = 8 and p = 0.1. Deter- mine the CDF and PMF of X and plot these.
Suppose Y is a discrete random variable with probability mass function p(y) - P(Y -y) -...
Suppose Y is a discrete random variable with probability mass function p(y) - P(Y -y) - fory - 1,2, ..., n. Show that p(y) satisfies the conditions of a pmf.
Problem 5. Let X be a binomial random variable with parameters n and p. Suppose that...
Problem 5. Let X be a binomial random variable with parameters n and p. Suppose that we want to generate a random variable Y whose probability mass function is the same as the conditional mass function of X given X-k, for some k-n. Let a = P(X-k), and suppose that the value of a has been computed (a) Give the inverse transform method for generating Y. (b) Give a second method for generating Y (c) For what values of a,...
Ciule jolh! PMF and the marginal PMFs? 6.14 Let X and Y be discrete random variables. Show that t...
ciule jolh! PMF and the marginal PMFs? 6.14 Let X and Y be discrete random variables. Show that the function p: R2 R defined by p(r, y) px(x)pr(y) is a joint PMF by verifying that it satisfies properties (a)-(c) of Proposition 6.1 on page 262. Hint: A subset of a countable set is countable CHAPTER SIX Joindy Discrete Random Variables 6.2 Joint and marginal PMFs of the discrete random variables x numher of bedrooms and momber of bwthrooms of a...
From textbook: "Let X have a negative binomial distribution with parameters r and p such that:...
From textbook:
"Let X have a negative binomial distribution with parameters r
and p such that:
Find E[X] and Var[X] without using the definition; instead,
consider how X can be written as a sum of independent random
variables."
Question: How do I do this?
p(k) = p' (1-p)*-, k=r,r+1,..
6. Imagine a negative binomial random variable X with p = 0.3 and r-3. (If you...
6. Imagine a negative binomial random variable X with p = 0.3 and r-3. (If you want to use R to determine the probabilities, be aware that the definition of a negative random variable is slightly different from the definition given in our text). Determine the following: a) E(X) b) V(X) c) P(X-20) d) P (X=19) e) P(X = 21)
If X is a nonnegative integer-valued random variable then the function P(z), defined for lzl s 1 by is called the probability generating function of X (a) Show that d* (b) With 0 being considered even, show that PX is even) = P(-1) + P(1) (c) If X is binomial with parameters n and p, show that Pix is even) -1+12p (d) If X is Poisson with mean A, show that 1+ e-24 2 P[X is even)- (e) If X...
The moment generating function ф(t) of random variable X is defined for all values of t by et*p(x), if X is discrete e f (x)dx, if X is continus (a) Find the moment generating function of a Binomial random variable X with parameters n (the total number of trials) and p (the probability of success). (b) If X and Y are independent Binomial random variables with parameters (n1 p) and (n2, p), respectively, then what is the distribution of X...
Problem 7 (15 points). Let X be random variable with the binomial distribution with parameters n and 0 <p<1. (1) Show that **- 1 = 2* for any 1 Sxsn. (2) Show that when 0 < x < (n + 1)p, P(X = x) is an increasing function x and for (n + 1)p <x Sn, P(X = x) is a decreasing function x. (3) A certain basketball player makes a foul shot with probability 0.80. Determine for whal value...
3. You are given a binomial random variable X, with parameters n = 8 and p = 0.1. Deter- mine the CDF and PMF of X and plot these.
Suppose Y is a discrete random variable with probability mass function p(y) - P(Y -y) - fory - 1,2, ..., n. Show that p(y) satisfies the conditions of a pmf.
Problem 5. Let X be a binomial random variable with parameters n and p. Suppose that we want to generate a random variable Y whose probability mass function is the same as the conditional mass function of X given X-k, for some k-n. Let a = P(X-k), and suppose that the value of a has been computed (a) Give the inverse transform method for generating Y. (b) Give a second method for generating Y (c) For what values of a,...
ciule jolh! PMF and the marginal PMFs? 6.14 Let X and Y be discrete random variables. Show that the function p: R2 R defined by p(r, y) px(x)pr(y) is a joint PMF by verifying that it satisfies properties (a)-(c) of Proposition 6.1 on page 262. Hint: A subset of a countable set is countable CHAPTER SIX Joindy Discrete Random Variables 6.2 Joint and marginal PMFs of the discrete random variables x numher of bedrooms and momber of bwthrooms of a...
From textbook:
"Let X have a negative binomial distribution with parameters r
and p such that:
Find E[X] and Var[X] without using the definition; instead,
consider how X can be written as a sum of independent random
variables."
Question: How do I do this?
p(k) = p' (1-p)*-, k=r,r+1,..
6. Imagine a negative binomial random variable X with p = 0.3 and r-3. (If you want to use R to determine the probabilities, be aware that the definition of a negative random variable is slightly different from the definition given in our text). Determine the following: a) E(X) b) V(X) c) P(X-20) d) P (X=19) e) P(X = 21)
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