Let X and Y be two discrete random variables with the given pmf pX (xi),i = 1, 2, . . . , n, and pY (yj ), j = 1, 2, . . . , m. Verify for following properties using thedefinition E[X] = ∑ni=1 xipX (xi).• (linearity) E[aX + bY + c] = aE[X] + bE[Y ] + c.• (monotonicity) If X ≤ Y , E[X] ≤ E[Y ].
Question
Let X and Y be two discrete random variables with the given pmf pX (xi),i = 1, 2, . . . , n, and pY (yj ), j = 1, 2, . . . , m. Verify for following properties using thedefinition E[X] = ∑ni=1 xipX (xi).• (linearity) E[aX + bY + c] = aE[X] + bE[Y ] + c.• (monotonicity) If X ≤ Y , E[X] ≤ E[Y ].
Solution
Sure, let's verify these properties:
- Linearity: E[aX + bY + c] = aE[X] + bE[Y] + c
The expected value of a random variable is the sum of the possible values of the variable each multiplied by the probability of that value occurring. So, we can write the expected value of aX + bY + c as:
E[aX + bY + c] = ∑ (axipX(xi) + byjpY(yj) + c*pX(xi)*pY(yj))
This can be separated into three sums:
= a*∑xipX(xi) + b∑yjpY(yj) + c∑pX(xi)*pY(yj)
The first sum is aE[X], the second sum is bE[Y], and the third sum is c (since the sum of the probabilities of all outcomes is 1). So, we have:
E[aX + bY + c] = aE[X] + bE[Y] + c
- Monotonicity: If X ≤ Y, E[X] ≤ E[Y]
If X is always less than or equal to Y, then for each possible value xi of X, there is a corresponding value yj of Y such that xi ≤ yj. Therefore, xipX(xi) ≤ yjpY(yj) for all i and j. Summing over all i and j, we get:
∑xipX(xi) ≤ ∑yjpY(yj)
which means E[X] ≤ E[Y]. So, the expected value of X is less than or equal to the expected value of Y.
Similar Questions
Let X and Y be the discrete random variables with the given pmf pX (xi)and pY (yi). Assume E[X2] < ∞, E[Y 2] < ∞. Variance and covariance are defined asV ar(X) = E[(X − E[X])2], Cov(X, Y ) = E[(X − E[X])(Y − E[Y ])]. Verify that• V ar(X) = E[X2] − E[X]2
(monotonicity) If X ≤ Y , E[X] ≤ E[Y ]
(Linearity of expectation II) Let X, Y be random variables and a, b, and c be constants. Use propertiesof integration/summation to show that:E(aX + bY + c) = aE[X] + bE[Y ] + c
1. Let X and Y be two random variables with joint pmf as follows.yfX,Y (x, y) -1 1 2x-2 0.15 0.10 0.051 0.25 0.25 0.20(a) Let event A = {{X is even} ∩ {Y is odd}}. Find P A. [2 marks](b) Find the marginal pmf’s fX (x) and fY (y). [3 marks](c) Find E(XY ). [2 marks](d) Find E(X + Y ). [2 marks](e) Are X and Y independent? Justify your answer. [1 mark](f) Let event A = {{X is even} ∩ {Y is odd}}. Compute the conditional probability massfunction fY |A(y) for Y given A occurs. [3 marks](g) Find the conditional pmf fX|Y (x|2). [2 marks](h) Find E[X|Y = 2]. [2 marks](i) Find Cov(X, Y ). [2 marks](j) Find Var(X + Y). [4 marks](k) Compute the correlation ρX,Y .
Context: The random variables X and Y have the joint PMF: px,y(x, y)=c*(x+y)^(2) if x belongs to {1,2,4} and y belongs to {1,3} and otherwise px,y(x,y) =0. Find the expectations E[XY].
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