1). Suppose that 1) p 1 P(X 0),i 1,2 P(Xi If p1p2, show that X1 >st X2.

Q: (-1)"2" /ñ Prove that if m is a positive interger, 1'(-ni +3) =1:3.5...(2m – 1)

A: I will use method of induction to prove the statment.

Q: 4.3.a Use Theorem 9.4 to show that Y is sufficient for p 4.3.b Find the MVUE for p

A:

Q: Let X - N(2, 4). Calculate P[X2 – 4X < 4].

A: P(X2-4X≤4)=0.7602

Q: Given that en 2= V and en 3=W,express each of the fallowing in terms of vand w a) in zv0/3

A:

Q: Let x, y e Z. Prove that if 3x+7y is odd or 5x + 6y is even, then x or y is even.

A: We have given that for x,y belongs to integers. If 3x+7y is odd or 5x+6y is even then x or y is even…

Q: The value(s) of x that satisfy In(x² + 1) – 2 In(x) = In(16) – In(8) are/is

A:

Q: If k >= 2 and x in R^ k prove that there exists y in R^ k such that y ne0 but xy = 0 Is this also…

A: If x has any components equal to 0, then y can be taken to have the corresponding components. equal…

Q: prove that 2. catX+c

A: We use that- sin2x + cos2x =1  so that 1-cos2x= sin2x

Q: Suppose that P(Xi = 1) =pi = 1-P(Xi = 0), i=1,2 If p1>=p2 , show that X1 >= X2.

A:

Q: Let x, y E R. Prove that if 21x + 35y = 45, then x and y are not both integers.

A:

Q: If Z ~ N(0,1), compute P(Z2 < 3)

A:

Q: 3 2 If A = use the Cayley-Hamilton theorem to determine A?. 5 1

A: We can solve this by using given information

Q: A. Let p(x) = 2x + 5x-3,m(x) = 2x- 1,and h(x) = Find: 1. (m-p)(x) 2. p(5)+ m(3)-h(1) +1 %3D x-2 3.…

A: A Given px=2x2+5x−3, mx=2x−1and hx=x+1x−2. 1 We have to find the m−px.…

Q: 5. Let r € R and A, = {q € Q : q < x} . Prove sup A, = x.

A: According to the given information, Consider the given set Ax:

Q: Prove that ((¬(p → r)) ≡ (r → p)) is a wff.

A: We have to solve given problem:

Q: HeW. By the Same way Prove that: X Ju (x) (X)

A:

Q: Let x, y EZ. Prove that 3|xy if and only if 3|x x or 3 y.

A:

Q: If m and n are positive integers and w = g(x"y"), then show that dw dw = my dy

A: Given w=g(xmyn).

Q: Show that if P(A;) = 1 for all i > 1, then P(N1 A¡) = 1.

A: We know that P(Ai) = 1 for i ≥1 Considering P(Ai), i=1,2,3,... to be independent events

Q: Prove that ifx > 3 and y> 4, then xy > 11.

A:

Q: 2. Show that p q is congruent to (p - 4)A .

A:

Q: Prove that i 224 s 2xP for xzO, y zo i and 1 t L-l with p>

A: Given that x,y≥0. And 1p+1q=1 with p&gt;1. Consider gx=xpp+1q-x. Then g'x=xp-1-1 and g'x&lt;0 when…

Q: 1. Given that r + px+q and 4x +q have a factor (x-r) where p,q and r are non-zero. Show that 4p…

A: topic - quadratic equations

Q: 4.51 Verify that BIBD with the parameter a = 8, r = 8, k = 4, and b = 16 does not exist.

A:

Q: Suppose that A ¥BSCN D and x E A. Use the method of "proof by contradiction" to show that if x e C,…

A: Given data : x∈Aandx∉C To show : x∈B

Q: Prove that if n> m - 1, then J,(x) = 2(x/2)"-m

A:

Q: Given that x, y, and z . Prove that if (x^2)|y and (y^3)|z, then (x^6)|z

A: Solution: The objective is to prove that the given statement

Q: b) Prove that if x <y and z<w, then x+z<y+w.

A: Given: Suppose x&lt;y and z≤w. To prove: x+z&lt;y+w.

Q: Prove by contradiction that if m, n € Z, then m² - n is even or m-n-1 is even.

A:

Q: *prove that for . P+2+3+... +n° = 2 n=!,2,3,

A:

Q: Theorem = 7 K 51 M 25

A:

Q: 1. Given: mZABC 80° |3D Prove: x = 8 Statements 6x+5 3x +3 Reasons MLABC 80 MLABD + MLDBC MLABC (3x+…

A:

Q: Let n E N coprime with p. Show that there exists x, y E Z coprime and such that p | x² + ny² is and…

A: To prove the given claim we have to show that :- x,y ∈ Z ; gcd(x,y)=1 ; p | x2 + ny2   implies…

Q: U is a neighborhood of x0 and U ⊂ V, The V is a neighborhood of x0. (2) If U1 ...., U2 are…

A: Solution ::

Q: 1. Do not use a theorem to find the ff. Z if yz? + z = 5xy – 1. %3D

A: In the multivariable function, if we have to find the partial derivative of one variable with…

Q: Prove that:  If 0 ≠ p ∈ Q and x ∉ Q, then (show that) a + x ∉ Q and ax ∉ q.

A: Given: Let a∈ℚ and x∉ℚ To show: a+x∉ℚax∉ℚ

Q: 1. Let A = {x E Z | 6 divides x} and B = {x € Z | 3 divides x}. Prove that A C B, and then prove…

A:

Q: 1. Use Leibnitz theorem and find P3(x)? . v = (x+ 1)°u (x – 1)

A:

Q: Supposernc N and n 4 0 Then show that rn is positive if and only if r is positive

A:

Q: show that if Z ~ N(0, 1) and Y ~x²(n) with Z and Y to be independent, then T: n

A:

Q: Let P= {p:p=6x-1, for some x EZ} and Q= {q:q=6y+5, for some y €Z}. Prove that P= Q

A: Answer

Q: What statement could be used to prove that m tn?

A:

Q: Theorem 5 there is auniqe Reul for X=2 Sit X ER

A:

Q: Prove (p a q) and (T→ (¬ (p →q))) are both wff:

A:

Q: If X ~ N(u =37,0 =7.2) What is the P(X <21) Find æ such thatP(X < æ) =0.38?

A:

Q: In Z, let H = &lt;5&gt; and K = &lt;7&gt;. Prove that Z = HK. Does Z = H X K?

A: Given: In Z H=5K=7

Q: Which theorem can be used to prove that ΔROM ΔΡΜΟ M P.

A: We can answer the question as below

Q: Under what conditions on A and B is (AB)-1 = A-1B-1? Prove your assertion

A: Assertion: If A and B are commutative, then  AB-1= A-1B-1.

Q: Show that yn Z, if n³+5 is odd then n is even

A:

Q: Let x, y e Z. Prove that if 3x + 7y is odd or 5x + 6y is even, then x or y is even.

A:

Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.