Math  /  Data & Statistics

QuestionQ1: For some event AA with P(A)=0.1P(A)=0.1 then P(AΩ)+P(ϕΩ)+P(ΩA)=P(A \mid \Omega)+P(\phi \mid \Omega)+P(\Omega \mid A)= A) 0.1 B) 1.2 C) 2.3 D) 1.1 E) None
Q2: Let XX be a random variable with E(X)=1E(X)=1 and E(X10+X)=2E\left(X^{10}+X\right)=2 Then E(X10)=E\left(X^{10}\right)= A) 0 B) 1 C) 2 D) 3 E) None
Q3: For x>0x>0 we have u(x)+3δ(y)=u(x)+3 \delta(y)= A) 1 B) 2 C) 3 D) 4 E) None
Q4: For RXY={(0,0),(1,1)}R_{X Y}=\{(0,0),(1,1)\}, if f(0,0)=0.2f(0,0)=0.2 and f(1,1)=0.8f(1,1)=0.8. Then E(XY)=E(X Y)= A) 1 B) 0.2 C) 0.8 D) 0.7 E) None
Q5: For some disjoint events A,BA, B with P(A)=0.2P(A)=0.2 and P(B)=0.4P(B)=0.4, we have P(AB)=P(A \cup B)= A) 0.2 B) 0.3 C) 0.4 D) 0.6 E) None
Q6: If P(A)=0.2P(A)=0.2 and P(AˉB)=P(BˉA)P(\bar{A} \cap B)=P(\bar{B} \cap A), then P(B)=P(B)= A) 0.1 B) 0.2 C) 0.4 D) 0.6 E) None
Q7: x3δ(x+1)dx=\int_{-\infty}^{\infty} x^{3} \delta(x+1) d x= A) -1 B) 8 C) -8 D) 1 E) None

Studdy Solution
Q7: Evaluate x3δ(x+1)dx\int_{-\infty}^{\infty} x^{3} \delta(x+1) \, dx.
Using the sifting property of the Dirac delta function: x3δ(x+1)dx=(1)3=1 \int_{-\infty}^{\infty} x^{3} \delta(x+1) \, dx = (-1)^{3} = -1
Answer: A) -1

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