Description
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#1. Consider the following joint probability distribution of X and Y |
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Y \ X |
1 |
2 |
3 |
4 |
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4 |
0 |
1/20 |
1/20 |
1/20 |
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3 |
1/20 |
2/20 |
3/20 |
1/20 |
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2 |
1/20 |
2/20 |
3/20 |
1/20 |
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1 |
1/20 |
1/20 |
1/20 |
0 |
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Find marginal distributions P(x) and P( y) .
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If Z X 2Y , find P(z) .
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Using b), find E(Z ) .
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Find E( X ) and E(Y ) then find E(Z ) using the expectation of X and expectation of Y
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Are X and Y independent?
#2. Let joint Probability distribution function of random variables X and Y be
1/ 3
1/ 3
P(x, y)
1/ 3
0
Are X and Y independent?
if (x, y) (1, 1)
if (x, y) (2, 0)
if (x, y) (0, 0)
otherwise .
#3. Each morning John eats some eggs. On any given morning, the number of eggs he eats is equally likely to 1, 2, 3, 4, or 5 independent of what he has done in the past. Let X be the number of eggs that John eats in 10 days. Find the mean and the variance of X.
#4. The time till failure of an electronic component has an Exponential distribution and it is known that 10% of components have failed by 1000 hours.
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What is the probability that a component is still working after 5000 hours?
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Find the mean and standard deviation of the time till failure.
#5. Let X be a continuous random variable with PDF,
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f X
ax2
: 0 x 2.
(x)
0
: otherwise
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Find a .
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Find variance of X.
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Find Cumulative Distribution Function (CDF) of X.
#6. Let X be a continuous random variable with PDF,
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f X
ke 2 x
:
x 0.
(x)
0
: otherwise
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Find k .
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Find mean of X.
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Find variance of X.
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Find CDF of X.
#7. Let X be a discrete random variable with probability distribution (probability mass function),
P(x) c (1/ 3) x , x 0, 1, 2,
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Find c such that P(x) is a legitimate PMF.
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Find Cumulative Distribution Function (cdf) of X, F(x) , x {0, 1, 2, }
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f X |
4x3 |
: |
0 x 1. |
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#8. Suppose X has probability density function |
(x) |
0 |
: otherwise |
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Find cdf of X.
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Using the answer in art a), find P(X 1/ 2) .
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Using the answer in art a), find P(1/ 3 X 2 / 3) .
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Suggested Problems
Chapter-3: 60-66, Chapter-5: 2,4,5,6,8,9,10,11.
