From a sack of fruit containing 3 oranges, 2 apples and 3 bananas, a random sample of 4 pieces of fruit is selected. If X is the number of oranges and is the number of apples in the sample, find
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![From a sack of fruit containing 3 oranges, 2 apples,
and 3 bananas, a random sample of 4 pieces of
fruit is selected. If X is the number of oranges and Y
is the number of apples in the sample, find
the joint probability distribution
formula of X and Y;
Your answer
Find P[(X, Y) E A], where A is
the region that is given by {(x, y)
|x + ys 2}. *
Your answer](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F81d89cc0-0f21-42b6-9c30-2b8e4a14e19e%2F969af036-569f-4462-aeb9-61de2d8be334%2Fkb9vsgq_processed.jpeg&w=3840&q=75)
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- From a sack of fruit containing 3 oranges, 2 apples, and 3 bananas, a random sample of 4 pieces of fruit is selected. If X is the number of oranges and Y is the number of apples in the sample, find (a) the joint probability distribution of X and Y ; (b) P [(X, Y ) ∈ A], where A is the region that is given by {(x, y) | x + y ≤ 2}.For any continuous random variables X, Y , Z and any constants a, b, show the following from the definition of the covariance:From a sack of fruit containing 3 bananas, 3 apples, and 3 oranges, a random sample of 4 pieces of fruit is selected. Suppose X is the number of bananas and Y is the number of apples in the sample. (a) Find the joint probability distribution of X and Y. (b) Find P[(X,Y)∈A], where A is the region that is given by {(x,y) | x+y≤3}.
- Find the moment-generating function of the contin-uous random variable X whose probability density is given by f(x) =1 for 0 < x < 10 elsewhere and use it to find μ1,μ2, and σ2.If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?If the probability density of X is given by f(x) =kx3(1 + 2x)6 for x > 00 elsewhere where k is an appropriate constant, find the probabilitydensity of the random variable Y = 2X 1 + 2X . Identify thedistribution of Y, and thus determine the value of k.
- Suppose that the probability density function of x is fx=3x2, 0<x<1 0, elsewhere Determine p(x < (1/3)), p((1/3) ≤ x < (2/3)), and p(x ≥ (2/3)) Determine the cumulative distribution function of x.Consider two independent random variables X1 andX2 having the same Cauchy distributionf(x) = 1π(1 + x2)for − q < x < qFind the probability density of Y1 = X1 + X2 by usingTheorem 1 to determine the joint probability density ofX1 and Y1 and then integrating out x1. Also, identify thedistribution of Y1.Given that the discrete random variables X has probability mass function, Pr(X=x)= (x/6 where x=1,2,3) 0 elsewhere, Describe and graph its cumulative distribution function,F(x)
- Verify that if T has a t distribution with ν degrees offreedom, then X = T2 has an F distribution with ν1 = 1and ν2 = ν degrees of freedom.Let X denote the voltage at the output of a microphone, and suppose that X has a uniform distribution on the interval from −1 to 1. The voltage is processed by a "hard limiter" with cutoff values −0.5 and 0.5, so the limiter output is a random variable Y related to X by Y = X if |X| ≤ 0.5, Y = 0.5 if X > 0.5, and Y = −0.5 if X < −0.5. (a) What is P(Y = 0.5)? (b) Obtain the cumulative distribution function of Y. Please show your work, thanks!Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0