1.) 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) EA], where A is the region that is given by {(x, y) |x +y<2}.
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- Assume that the probability that an airplane engine will fail during a torture test is 12and that the aircraft in question has 4 engines. Construct a sample space for the torture test. Use S for survive and F for fail.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}.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
- If a random variable X has a discrete uniform distribution. fx(x)=1/k for x=1,2,..,k;0 otherwise. Derive P.G.F of X and compute E(2x+1)In a box of gadgets containing 3 Samsung phones, 2 iPhones, and 3 Google Pixelphones, a random sample of 4 phones is selected. If X is the number of Samsung phonesand Y is the number of iPhones in the sample, a. the joint probability distribution table of X and Y;b. P[(X, Y ) ∈ A], where A is the region that is given by {(x, y) | x + y ≤ 2}.c. f(y|x=2) for all values of y (tabulate);d. P(X = 0 | Y = 2).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}.
- If the random variable X and Y are independent of each other. X~N(0,1) and the probabilitymass function of Y is P(Y=0)P(Y=1)=1/2. If Fz(z) denotes the cumulative distribution functionof random variable Z=XY, then the number of discontinuites of function Fz(z) isSuppose that X is a discrete random variable with P(X = 0) = 0.3, P(X = 1) = 0.2, P(X =2) = 0.15, and P(X = 3) = 0.35. Graph the frequency function and the cumulative distribution function of X.In a box of gadgets containing 3 Samsung phones, 2 iPhones, and 3 Google Pixel phones, a random sample of 4 phones is selected. If X is the number of Samsung phones and Y is the number of iPhones in the sample, find: a. the joint probability distribution table of X and Y;b. P[(X, Y ) ∈ A], where A is the region that is given by {(x, y) | x + y ≤ 2}.c. f(y|x=2) for all values of y (tabulate);d. P(X = 0 | Y = 2).
- (a) Let F denote the cumulative distribution function (cdf) of a uniformly distributed random variable X. If F(2) = 0.3, what is the probability that X is greater than 2 ? (b) Let F denote the cdf of a uniformly distributed random variable X. If F(2) = 0.3, and F(3) = 0.6, what is F(6) ? (c) Suppose X and Y are Poisson Random Variables. X has a mean of 1 and Y has a mean of 2. X and Y are correlated with CORR (X,Y)=0.5. whats the variance of X+Y8.- Given that X is a hypergeometric random variable with N = 10, n = 5, and r = 7: a. Display the probability distribution for X in tabular form. b. Compute the mean and variance of X. c. Graph p(x) and locate μ and the interval μ ±2σ on the graph. d. What is the probability that x will fall within the interval μ ±2σ ?